Algebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions
|
|
- Sarah Cameron
- 5 years ago
- Views:
Transcription
1 Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector Unit Circle Graphing Trigonometric Identities
2 Slide 4 / 162 Radians & Degrees and Co-Terminal Angles Return to Table of Contents A few definitions: A central angle of a circle is an angle whose vertex is the center of the circle. An intercepted arc is the part of the circle that includes the points of intersection with the central angle and all the points in the interior of the angle. Slide 5 / 162 intercepted arc central angle Radians and Degrees Slide 6 / 162 One radian is the measure of a central angle that intercepts an arc whose length is equal to the radius of the circle. There are, or a little more than 6, radians in a circle. Click on the circle for an animated view of radians.
3 Converting from Degrees to Radians Slide 7 / 162 There are 360 in a circle. Therefore 360 = 2 radians 2 1 = 360 = 180 radians Use this conversion factor to covert degrees to radians. Example: Convert 50 and 90 to radians. 50 = 5 radians = radians Converting from Radians to Degrees Slide 8 / radians = radian = = 180 degrees 2 Use this conversion factor to covert radians to degrees. Example: Convert and to radians = = 180 Converting between Radians and Degrees Slide 9 / 162 Convert degrees to radians
4 Converting between Radians and Degrees Slide 10 / 162 Convert radians to degrees radians radians radians Slide 11 / 162 Slide 12 / 162
5 Slide 13 / 162 Slide 14 / Convert radians to degrees: Angles Slide 15 / 162 Terminal side Initial side Terminal side Initial side Angle Angle in standard position An angle is formed by rotating a ray about its endpoint. The starting position is the initial side and the ending position is the terminal side. When, on the coordinate plane, the vertex of the angle is the origin and the initial side is the positive x-axis, the angle is in standard position.
6 Positive Angle - terminal side rotates in a counterclockwise direction Negative Angle - terminal side rotates in a clockwise direction Slide 16 / 162 α = - 37 Drawing angles in standard position Slide 17 / Each quadrant is 90, and 310 is 40 more than 270, so the terminal side is 40 past the negative y-axis. 500 is 140 more than 360, so the angle makes a complete revolution counterclockwise and then another 140. Coterminal Angles Slide 18 / 162 Angles that have the same terminating side are coterminal. To find coterminal angles add or subtract multiples of 360 for degrees and 2 for radians. Example: Find one positive and one negative angle that are terminal with = =
7 5 Which angles are coterminal with 40? (Select all that are correct.) A 320 B -320 C 400 D -400 Slide 19 / Which graph represents 425? Slide 20 / 162 A B C D 7 Which graph represents? Slide 21 / 162 A B C D
8 8 Which angle is NOT coterminal with -55? Slide 22 / 162 A 305 B 665 C -415 D Which angle is coterminal with? Slide 23 / 162 A B C D Slide 24 / 162 Arc Length & Area of a Sector Return to Table of Contents
9 Slide 25 / 162 Arc length and the area of a sector (Measured in radians) r arc length s sector Arc length: s = r Area of sector: A = How do these formulas relate to the area and the circumference of a circle? Who is getting more pie? Who is getting more of the crust at the outer edge? Slide 26 / Emily's slice is cut from a 9 inch pie. Chester's slice is cut from an 8 inch pie. (Assume both pies are the same height.) (Try to work this out in your groups. The solution is on the next slide) Slide 27 / click The top of Emily's piece has an area of click The top of Chester's piece has an area of Emily's crust has a length of Chester's crust has a length of
10 10 What is the top surface area of this slice of pizza from an 18-inch pie? Slide 28 / What is the arc length of the outer edge of this slice of pizza from an 18-inch pie? Slide 29 / If the radius of this circular saw blade is 10 inches and there are 40 teeth on the blade, how far apart are the tips of the teeth? Slide 30 / 162
11 13 Challenge Question: Given a dart board as shown. If a dart thrown randomly lands somewhere on the board, what is the probability that it will land on a red region? Slide 31 / in 8 inches Slide 32 / 162 Unit Circle Return to Table of Contents The Unit Circle Slide 33 / 162 The circle x 2 + y 2 = 1, with center (0,0) and radius 1, is called the unit circle. Quadrant II: x is negative and y is positive (0,1) 1 Quadrant I: x and y are both positive (-1,0) Quadrant III: x and y are both negative (0,-1) (1,0) Quadrant IV: x is positive and y is negative
12 The unit circle allows us to extend trigonometry beyond angles of triangles to angles of all measures. (-1,0) (0,-1) (0,1) 1 θ a (a,b) b (1,0) In this triangle, b sin#= 1 = b a cos# = 1 = a so the coordinates of (a,b) are also (cos#, sin#) Slide 34 / 162 For any angle in standard position, the point where the terminal side of the angle intercepts the circle is called the terminal point. In this example, the terminal point is in Quadrant IV If we look at the triangle, we can see that sin(-55 ) = 0.82 cos(-55 ) = 0.57 EXCEPT that we have to take the direction into account, and so sin(-55 ) is negative because the y value is below the x-axis. Slide 35 / 162 For any angle θ in standard position, the terminal point has coordinates (cosθ, sinθ). Slide 36 / 162 Click the star below to go to the Khan Academy Unit Circle Manipulative try some problems:
13 What are the coordinates of point C? Slide 37 / 162 In this example, we know the angle. Using a calculator, we find that cos and sin 44.69, so the coordinates of C are approximately (0.72, 0.69). 1 Note that ! The Tangent Function Recall SOH-CAH-TOA Slide 38 / 162 sin # = opp hyp cos # = adj hyp opp tan # = adj opposite side hypotenuse # adjacent side It is also true that tan = sin # cos. # Why? opp hyp = opp hyp = opp adj hyp adj adj = tan # hyp Angles in the Unit Circle Slide 39 / 162 Example: Given a terminal point on the unit circle (- ). Find the value of cos, sin and tan of the angle. Solution: Let the angle be. x = cos, so cos =. y = sin, so sin =. tan = = = = (Shortcut: Just cross out the 41's in the complex fraction.)
14 Example: Given a terminal point csc#. Note the "hidden" Pythagorean Triple, 8, 15, 17)., find #, tan# and To find #, use sin -1 or cos -1 : sin -1 ( ) = # # # 28.1 tan# = sin#/ cos# tan # = csc# = 1/ sin# csc # = Slide 40 / 162 Example: Find the x-value of point A, θ and the tan θ. For every point on the circle, Slide 41 / (, - 13 ) θ A Since x is in quadrant III, x = sin -1 5 (- 13 ) -22.3, BUT θ is in quadrant III, so θ = = (notice how and have the same sine) sin θ tan θ = cos θ = = 5 12 Example: Given the terminal point of ( -5 / 13, -12 / 13). Find sin x, cos x, and tan x. Slide 42 / 162
15 14 What is tan θ? 3 (- 5, ) Slide 43 / 162 A θ B C D 15 What is sin θ? 3 (- 5, ) Slide 44 / 162 A θ B C D 16 What is θ (give your answer to the nearest degree)? Slide 45 / (- 5, ) θ
16 17 Given the terminal point, find tan x. Slide 46 / 162 Slide 47 / 162 Equilateral and isosceles triangles occur frequently in geometry and trigonometry. The angles in these triangles are multiples of 30 and 45. A calculator will give approximate values for the trig functions of these angles, but we often need to know the exact values. Slide 48 / 162 Isosceles Right Triangle Equilateral Triangle (the altitude divides the triangle into two triangles)
17 Special Right Triangles Slide 49 / 162 (see Triangle Trig Review unit for more detail on this topic) Special Triangles and the Unit Circle Slide 50 / 162 (-, ) (, ) Multiples of 45 angles have sin and cos of ±, depending on the quadrant. 30 o 45 o 60 o 60 o Slide 51 / o 30 o 30 o 30 o 45 o 60 o 60 o 45 o
18 Drag the degree and radian angle measures to the angles of the circle: # 5# # 3# 7# 3# 0 # 2# Slide 52 / Fill in the coordinates of x and y for each point on the unit circle: Slide 53 / 162 (, ) 3# 4 (, ) # 2 # 4 (, ) (, ) # (, ) 5# 4 3# 2 7# 4 2# 0 (, ) (, ) (, ) Special Triangles and the Unit Circle Slide 54 / 162 (, ) 1 (, ) Angles that are multiples of 30 have sin and cos of ± and ±.
19 Drag the degree and radian angle measures to the angles of the circle: 5# # # # 3# 0 2# 4# # 2# 7# 11# 5# Slide 55 / Drag in the coordinates of x and y for each point on the unit circle: Slide 56 / 162 (, ) (, ) (, ) 2# 3 # 5# 6 7# (, ) 6 4# 3 (, ) # 2 (, ) (, ) # 3 (, ) # 6 2# 0 (, ) 11# (, ) 6 3# 5# 2 3 (, ) (, ) Special Angles in Degrees Slide 57 / 162
20 Radian Values of Special Angles Slide 58 / 162 Exact Values of Special Angles Slide 59 / 162 Put it all together... Slide 60 / 162
21 Exact values of special angles Slide 61 / 162 Complete the table below: Degrees Radians sin θ cos θ tan θ Slide 62 / 162 Slide 63 / 162
22 Slide 64 / 162 If we know one trig function value and the quadrant in which the angle lies, we can find the angle and the other trig values. Example: If tan =, and sin < 0, find sin, cos and the value of. Slide 65 / 162 Solution: Since tan is positive and sin is negative, the terminal side of must be in Quadrant III. Draw a right triangle in Quadrant III. Use the Pythagorean Theorem to find the length of the hypotenuse: opp -3 adj hyp (Continued on next slide) Once we know the lengths for each side, we can calculate the sin, cos and the angle. Used the signed numbers to get the correct values. opp -3 adj hyp Slide 66 / 162 sin = = cos = = Use any inverse trig function to find the angle. tan-1( ) Because the angle is in QIII, we need to add = 216.7, so 217.
23 Slide 67 / 162 Slide 68 / 162 Slide 69 / 162
24 Slide 70 / Which functions are positive in the second quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Slide 71 / Which functions are positive in the fourth quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x Slide 72 / Which functions are positive in the third quadrant? Choose all that apply. A B C D E F cos x sin x tan x sec x csc x cot x
25 Slide 73 / 162 Graphing Trig Functions Return to Table of Contents If you have Geogebra available on your computer, click the star below to download a geogebratube animated graph of the trig functions: Slide 74 / 162 (Once the webpage opens, click on Download) Graphing the Sine Function, y = sin x Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of, and the corresponding values of y or sin x. Slide 75 / 162 (Remember, is just a bit more than 3.) Since the values are based on a circle, values will repeat.
26 Notice that the graph of y = sin x increases from 0 to 1, then decreases back to 0 and then to -1 and then goes back up to 0, as x increases from 0 to 2. Slide 76 / 162 Graphing the Cosine Curve Slide 77 / 162 Make a table of values just as we did for sin. We could use any interval, but are choosing from 0 to 2. Since the values are based on a circle, values will repeat. Notice that the graph of y = cos x starts at 1, decreases to -1 and then goes back up to 1 as x increases from 0 to 2. Slide 78 / 162
27 Compare the graphs: Slide 79 / 162 y = sin x y = cos x How are they similar and how are they different? Characteristics of y = sin x and y = cos x Slide 80 / 162 range: -1 y 1 amplitude = 1 period = 2 Domain: set of real numbers (x can be anything) Range: -1 y 1 Amplitude: one-half the distance from the minimum of the graph to the maximum or 1. The functions are periodic - the pattern repeats every 2 units. Predict, Explore, Confirm Slide 81 / Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?
28 y = a sin x or y = a cos x Slide 82 / 162 Amplitude is a positive number that represents one-half the difference between the minimum and the maximum values, or the distance from the midline to the maximum. Consider the graphs of y = sin x What do you notice about these y = 2sin x graphs? What does the value of y = sin x "a" do to the graph? Slide 83 / 162 y = 2sin x y = sin x y = sin x Name the amplitude of each graph. As shown in the graph below, the graph of y = -3cos xis a reflection over the x-axis of the graph of y = 3cos x. What is the amplitude of each function? y = 3cos x y = -3cos x Slide 84 / 162 The domain of each function is the set of real numbers and the range is {x -3 x 3}.
29 Sketch each graph on the interval from 0 to 2 : Slide 85 / 162 y = 4cos x y = -.25 sin x Slide 86 / What is the amplitude of y = 3cos x? Slide 87 / What is the amplitude of y = 0.25cos x?
30 Slide 88 / What is the amplitude of y = -sin x? 28 What is the range of the function y = 2sin x? Slide 89 / 162 A All real numbers B -2 < x < 2 C 0 x 2 D -2 x 2 29 What is the domain of y = -3cos x? Slide 90 / 162 A All real numbers B -3 < x < 3 C 0 x 3 D -3 x 3
31 30 Which graph represents the function y = -2sin x? Slide 91 / 162 A B C D 31 What is the amplitude of the graph below? Slide 92 / 162 Predict, Explore, Confirm Slide 93 / Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?
32 Slide 94 / 162 A periodic function is one that repeats its values at regular intervals. One complete repetition of the pattern is called a cycle. The period is the length of one complete cycle. The trig functions are periodic functions. The basic sine and cosine curves have a period of 2, meaning that the graph completes one complete cycle in 2 units. y = sin bx or y = cos bx Slide 95 / 162 Consider the graphs of y = cos xand y = cos 2x. y = cos x one cycle y = cos 2x Notice that the graph of y = cos 2xcompletes one cycle twice as fast, or in units. y = cos x completes 1 cycle in 2#. So the period is 2π. Slide 96 / 162 y = cos 2x completes 2 cycles in 2# or 1 cycle in #. The period is #. y = cos 0.5x completes a cycle in 4#. The period is 4#.
33 Slide 97 / 162 The period for y = cos bx or y = sin bx is P = 2 b y = cos x b = 1 2 P = 1 = 2 y = cos 2x b = 2 P = 2 2 = y = cos 0.5x b = 0.5 P = = 4 Slide 98 / What is the period of A B C D 33 What is the period of Slide 99 / 162 A B C D
34 Slide 100 / What is the period of A B C D Sketch the graph of each function from x = 0 to x = 2. Slide 101 / 162 y = 2cos 3x y = cos x y = sin 2x y = -2cos 2x 35 What is the period of the graph below? Slide 102 / 162 A B 2 C 3 2 D 2
35 Slide 103 / What is the period of the graph shown? A B C D Slide 104 / What is the equation of this function? A B C D y = sin 3x y = cos 3x y = 3cos x y = 3sin x Predict, Explore, Confirm Slide 105 / Using your prior knowledge of transforming functions, predict what happens to the following functions: 2. Using your graphing calculator, insert the parent function into and the transformed function into. Compare the graphs. 3. Do your conclusions match your predictions?
36 Translating Sine and Cosine Functions Slide 106 / 162 Trig functions can be translated in the same way as any other function. The horizontal shift is called a phase shift. What are your conclusions from the graphing calculator activity? Drag each equation to the matching graph Slide 107 / 162 Horizontal or phase shift y = cos x y = cos (x + ) 2 Vertical shift y = sin x y = sin x + 2 k Slide 108 / 162
37 Slide 109 / 162 Consider the graphs of Slide 110 / 162 and (which is which?) In order to determine the phase shift, the coefficient of x must be factored out. In shift is. In, the 2 is factored out. The phase, when the 2 is factored out, we get. The phase shift is. Slide 111 / 162 Another way to determine the phase shift is to set the quantity inside the parenthesis equal to 0 and solve for the variable. Example: Set Solve for x: So, the phase shift is 2.
38 Slide 112 / 162 Slide 113 / 162 Vertical Shift y= sin (x) + k or y= cos (x) + k The k moves the graph up or down. The graph below is of the equation y = 2 sin (3x). The midline of this graph is the horizontal line y = 0. Sketch the graph of y = 2 sin (3x) + 1. Slide 114 / 162
39 Slide 115 / What is the vertical shift in Slide 116 / What is the vertical shift in Slide 117 / What is the vertical shift in
40 Slide 118 / 162 Graphing a Sine or Cosine Function: Slide 119 / 162 Step 1: Identify the amplitude, period, phase shift and vertical shift. Step 2: Draw the midline (y = k) Step 3: Find 5 key points - maximums, minimums and points on the midline Step 4: Draw the graph through the 5 points. Example: Slide 120 / 162 Step 1: Amplitude: -1 = 1 Period: Phase Shift: Vertical Shift: 2 (up 2)
41 Step 2: Draw the midline y = 2 Step 3: Find the 5 key points Slide 121 / 162 Note: for x, adding the cycle, 3 by 4. comes from dividing For y, adding and subtracting 1 comes from the amplitude. Slide 122 / 162 Step 4: Graph You try: Slide 123 / 162
42 Slide 124 / 162 Slide 125 / 162 Slide 126 / 162
43 Slide 127 / 162 Slide 128 / What is the amplitude of this cosine graph? Slide 129 / What is the period of this cosine graph? (use 3.14 for pi)
44 Slide 130 / What is the vertical shift of this cosine graph? Slide 131 / Which of the following of the following is an equation for the graph? A B C D The equation y = 4.2cos (π/6(x - 1)) can be used to model the average temperature of Wellington, NZ in degrees Celsius, where x represents the month, Sketch the graph of this equation. What is the average temperature in June? Slide 132 / 162
45 Graphing the Tangent Function Graph by creating a table of values of key points. One option is to use the set of values for x that are multiples of, and the corresponding values of y or tan x. Slide 133 / 162 Notice that the tangent function is not defined for values of x where cos x = 0, or starting at and every units in either direction. Slide 134 / 162 This is shown on the graph by the vertical lines, or asymptotes at these x values. The period of the function is units, because there is one complete cycle from to. As x approaches or any odd multiple of from the left, y increases and approaches infinity. As x approaches from the right, y decreases and approaches negative infinity. Example: Sketch the graph of y = tan (x + ) + 2 Slide 135 / 162 Asymptotes will be at 0,, 2, etc. The midline will be at y = 2.
46 53 Which graph represents y = -tan x? Slide 136 / 162 A B C D Slide 137 / 162 Trigonometric Identities Return to Table of Contents Key Ideas An identity is a mathematical equation that is true for all defined values of the variable. A trigonometric identity is an identity that contains one or more trig ratios. By contrast, a conditional equation is one that is only true for a limited set of values. By learning trig identities, we will be able to replace complicated expressions with simpler ones to solve and verify more difficult equations and identities. Slide 138 / 162
47 Drag each equation into the correct box: Slide 139 / 162 3x + 4 = 3x + 4 3x + 4 = 9 5x - 7y = -(7y - 5x) 2x 5 =x 3 sin # + cos # = 1 tan θ cot θ =1 2(x-1) = 2x - 2 (x + 3) 2 = x sin 4x = 4sin x Identities Conditional Equations Basic Trig Identities Slide 140 / 162 Reciprocal Identities csc # = 1 sin # sin # = 1 csc # sec # = 1 cos # cos # = 1 sec # cot # = 1 tan # tan # = 1 cot # Tangent Identity tan # = sin # cos # Cotangent Identity cot # = cos # sin # Slide 141 / 162 By using the basic identities we can change an expression into an equivalent expression. Also, all of the rules of addition, subtraction, multiplication and division that we learned to solve equations and manipulate expressions can be applied to trig expressions and equations.
48 Slide 142 / 162 Algebraic example Trig example (x - y)(x + y) = x 2 - y 2 (1 - cos #)(1 + cos #) = 1 - cos 2 # Pythagorean Identities Slide 143 / 162 Recall the unit circle, x 2 + y 2 = 1. (-1,0) (0,1) 1 (0,-1) (cos #,sin #) (1,0) For any point (x, y) on the circle, its coordinates are (cos #, sin #). Therefore, (cos #) 2 + (sin #) 2 = 1 2, which is usually written as cos 2 θ + sin 2 θ = 1 Slide 144 / 162 Pythagorean Identities How do we transform the first identity, which is derived from the unit circle, to the other two?
49 Alternative Forms of Identities Slide 145 / 162 Since we know that = 8, we also know that 8-5 = 3 and 8-3 = 5. In elementary school we call these equivalent equations "fact families". Similarly, if cos 2 θ + sin 2 θ = 1, it follows that 1 - cos 2 θ = sin 2 θ and 1 - sin 2 θ = cos 2 θ. More Alternative Forms Slide 146 / 162 Another fact family tells that since follows that 4 5 = = 4, it 1 Since sec θ = cos θ, then sec θ cos θ = 1 (multiply both sides of the first equation by cos #). Simplifying Trig Expressions Slide 147 / 162 Example 1: Simplify csc θ cos θ tan θ. Rewrite each trig ratio in terms of cos and sin: 1 sin # sin θ cos θ cos # = 1 Example 2: Simplify csc 2 θ(1 - cos 2 θ). 1 sin 2 θ (sin2 θ) = 1 (When multiplying fractions, it is often easier to reduce or cancel before you multiply.)
50 Verifying an Identity Transform one side of the identity to be the same as the other side Slide 148 / 162 Example 1: Verify sin # cot # = cos # sin # cos # = cos # sin # Example 2: Verify cos θ csc θ tan θ = 1 cos # 1 sin # = 1 sin # cos # Simplify: Slide 149 / 162 Simplify: Slide 150 / 162
51 Simplify: Slide 151 / 162 Slide 152 / 162 Slide 153 / 162 Simplify:
52 Verify: Slide 154 / 162 Slide 155 / 162 Verify: Slide 156 / 162
53 Slide 157 / 162 Slide 158 / Which equation is NOT an identity? Slide 159 / 162 A sin 2 x= 1 - cos 2 x B 2 cot x = 2cos x sin x C tan 2 x = sec 2 x - 1 D sin 2 x = cos 2 x - 1
54 55 The following expression can be simplified to which choice? Slide 160 / 162 A B C D 56 The following expression can be simplified to which choice? Slide 161 / 162 A B C D 57 The following expression can be simplified to which choice? Slide 162 / 162 A B C D
Algebra II Trigonometric Functions
Slide 1 / 162 Slide 2 / 162 Algebra II Trigonometric Functions 2015-12-17 www.njctl.org Slide 3 / 162 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc
More informationA lg e b ra II. Trig o n o m e tric F u n c tio
1 A lg e b ra II Trig o n o m e tric F u n c tio 2015-12-17 www.njctl.org 2 Trig Functions click on the topic to go to that section Radians & Degrees & Co-terminal angles Arc Length & Area of a Sector
More informationA trigonometric ratio is a,
ALGEBRA II Chapter 13 Notes The word trigonometry is derived from the ancient Greek language and means measurement of triangles. Section 13.1 Right-Triangle Trigonometry Objectives: 1. Find the trigonometric
More informationUnit 13: Periodic Functions and Trig
Date Period Unit 13: Periodic Functions and Trig Day Topic 0 Special Right Triangles and Periodic Function 1 Special Right Triangles Standard Position Coterminal Angles 2 Unit Circle Cosine & Sine (x,
More information4.1: Angles & Angle Measure
4.1: Angles & Angle Measure In Trigonometry, we use degrees to measure angles in triangles. However, degree is not user friendly in many situations (just as % is not user friendly unless we change it into
More informationChapter 4: Trigonometry
Chapter 4: Trigonometry Section 4-1: Radian and Degree Measure INTRODUCTION An angle is determined by rotating a ray about its endpoint. The starting position of the ray is the of the angle, and the position
More informationUnit 7: Trigonometry Part 1
100 Unit 7: Trigonometry Part 1 Right Triangle Trigonometry Hypotenuse a) Sine sin( α ) = d) Cosecant csc( α ) = α Adjacent Opposite b) Cosine cos( α ) = e) Secant sec( α ) = c) Tangent f) Cotangent tan(
More informationCommon Core Standards Addressed in this Resource
Common Core Standards Addressed in this Resource N-CN.4 - Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular
More informationUnit Circle. Project Response Sheet
NAME: PROJECT ACTIVITY: Trigonometry TOPIC Unit Circle GOALS MATERIALS Explore Degree and Radian Measure Explore x- and y- coordinates on the Unit Circle Investigate Odd and Even functions Investigate
More information: Find the values of the six trigonometric functions for θ. Special Right Triangles:
ALGEBRA 2 CHAPTER 13 NOTES Section 13-1 Right Triangle Trig Understand and use trigonometric relationships of acute angles in triangles. 12.F.TF.3 CC.9- Determine side lengths of right triangles by using
More informationPre-calculus Chapter 4 Part 1 NAME: P.
Pre-calculus NAME: P. Date Day Lesson Assigned Due 2/12 Tuesday 4.3 Pg. 284: Vocab: 1-3. Ex: 1, 2, 7-13, 27-32, 43, 44, 47 a-c, 57, 58, 63-66 (degrees only), 69, 72, 74, 75, 78, 79, 81, 82, 86, 90, 94,
More informationUnit 2 Intro to Angles and Trigonometry
HARTFIELD PRECALCULUS UNIT 2 NOTES PAGE 1 Unit 2 Intro to Angles and Trigonometry This is a BASIC CALCULATORS ONLY unit. (2) Definition of an Angle (3) Angle Measurements & Notation (4) Conversions of
More informationAppendix D Trigonometry
Math 151 c Lynch 1 of 8 Appendix D Trigonometry Definition. Angles can be measure in either degree or radians with one complete revolution 360 or 2 rad. Then Example 1. rad = 180 (a) Convert 3 4 into degrees.
More informationTrigonometry Review Day 1
Name Trigonometry Review Day 1 Algebra II Rotations and Angle Terminology II Terminal y I Positive angles rotate in a counterclockwise direction. Reference Ray Negative angles rotate in a clockwise direction.
More informationSection 5: Introduction to Trigonometry and Graphs
Section 5: Introduction to Trigonometry and Graphs The following maps the videos in this section to the Texas Essential Knowledge and Skills for Mathematics TAC 111.42(c). 5.01 Radians and Degree Measurements
More informationCCNY Math Review Chapters 5 and 6: Trigonometric functions and graphs
Ch 5. Trigonometry 6. Angles 6. Right triangles 6. Trig funs for general angles 5.: Trigonometric functions and graphs 5.5 Inverse functions CCNY Math Review Chapters 5 and 6: Trigonometric functions and
More informationPrecalculus: Graphs of Tangent, Cotangent, Secant, and Cosecant Practice Problems. Questions
Questions 1. Describe the graph of the function in terms of basic trigonometric functions. Locate the vertical asymptotes and sketch two periods of the function. y = 3 tan(x/2) 2. Solve the equation csc
More informationReview of Trigonometry
Worksheet 8 Properties of Trigonometric Functions Section Review of Trigonometry This section reviews some of the material covered in Worksheets 8, and The reader should be familiar with the trig ratios,
More informationAlgebra II. Slide 1 / 92. Slide 2 / 92. Slide 3 / 92. Trigonometry of the Triangle. Trig Functions
Slide 1 / 92 Algebra II Slide 2 / 92 Trigonometry of the Triangle 2015-04-21 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 92 Trigonometry of the Right Triangle Inverse
More informationSNAP Centre Workshop. Introduction to Trigonometry
SNAP Centre Workshop Introduction to Trigonometry 62 Right Triangle Review A right triangle is any triangle that contains a 90 degree angle. There are six pieces of information we can know about a given
More informationAlgebra II. Chapter 13 Notes Sections 13.1 & 13.2
Algebra II Chapter 13 Notes Sections 13.1 & 13.2 Name Algebra II 13.1 Right Triangle Trigonometry Day One Today I am using SOHCAHTOA and special right triangle to solve trig problems. I am successful
More informationMATHEMATICS 105 Plane Trigonometry
Chapter I THE TRIGONOMETRIC FUNCTIONS MATHEMATICS 105 Plane Trigonometry INTRODUCTION The word trigonometry literally means triangle measurement. It is concerned with the measurement of the parts, sides,
More informationGanado Unified School District Pre-Calculus 11 th /12 th Grade
Ganado Unified School District Pre-Calculus 11 th /12 th Grade PACING Guide SY 2016-2017 Timeline & Resources Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to highlight
More informationTrigonometry. 9.1 Radian and Degree Measure
Trigonometry 9.1 Radian and Degree Measure Angle Measures I am aware of three ways to measure angles: degrees, radians, and gradians. In all cases, an angle in standard position has its vertex at the origin,
More informationby Kevin M. Chevalier
Precalculus Review Handout.4 Trigonometric Functions: Identities, Graphs, and Equations, Part I by Kevin M. Chevalier Angles, Degree and Radian Measures An angle is composed of: an initial ray (side) -
More informationsin30 = sin60 = cos30 = cos60 = tan30 = tan60 =
Precalculus Notes Trig-Day 1 x Right Triangle 5 How do we find the hypotenuse? 1 sinθ = cosθ = tanθ = Reciprocals: Hint: Every function pair has a co in it. sinθ = cscθ = sinθ = cscθ = cosθ = secθ = cosθ
More informationMath-3 Lesson 6-1. Trigonometric Ratios for Right Triangles and Extending to Obtuse angles.
Math-3 Lesson 6-1 Trigonometric Ratios for Right Triangles and Extending to Obtuse angles. Right Triangle: has one angle whose measure is. 90 The short sides of the triangle are called legs. The side osite
More informationA lg e b ra II. Trig o n o m e try o f th e Tria n g le
1 A lg e b ra II Trig o n o m e try o f th e Tria n g le 2015-04-21 www.njctl.org 2 Trig Functions click on the topic to go to that section Trigonometry of the Right Triangle Inverse Trig Functions Problem
More informationUnit 2: Trigonometry. This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses.
Unit 2: Trigonometry This lesson is not covered in your workbook. It is a review of trigonometry topics from previous courses. Pythagorean Theorem Recall that, for any right angled triangle, the square
More informationGanado Unified School District Trigonometry/Pre-Calculus 12 th Grade
Ganado Unified School District Trigonometry/Pre-Calculus 12 th Grade PACING Guide SY 2014-2015 Timeline & Resources Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to
More informationTrigonometric Ratios and Functions
Algebra 2/Trig Unit 8 Notes Packet Name: Date: Period: # Trigonometric Ratios and Functions (1) Worksheet (Pythagorean Theorem and Special Right Triangles) (2) Worksheet (Special Right Triangles) (3) Page
More informationMAC Learning Objectives. Learning Objectives (Cont.) Module 2 Acute Angles and Right Triangles
MAC 1114 Module 2 Acute Angles and Right Triangles Learning Objectives Upon completing this module, you should be able to: 1. Express the trigonometric ratios in terms of the sides of the triangle given
More informationAP Calculus Summer Review Packet
AP Calculus Summer Review Packet Name: Date began: Completed: **A Formula Sheet has been stapled to the back for your convenience!** Email anytime with questions: danna.seigle@henry.k1.ga.us Complex Fractions
More informationGanado Unified School District #20 (Pre-Calculus 11th/12th Grade)
Ganado Unified School District #20 (Pre-Calculus 11th/12th Grade) PACING Guide SY 2018-2019 Timeline & Quarter 1 AZ College and Career Readiness Standard HS.A-CED.4. Rearrange formulas to highlight a quantity
More informationTrigonometry and the Unit Circle. Chapter 4
Trigonometry and the Unit Circle Chapter 4 Topics Demonstrate an understanding of angles in standard position, expressed in degrees and radians. Develop and apply the equation of the unit circle. Solve
More information5.1 Angles & Their Measures. Measurement of angle is amount of rotation from initial side to terminal side. radians = 60 degrees
.1 Angles & Their Measures An angle is determined by rotating array at its endpoint. Starting side is initial ending side is terminal Endpoint of ray is the vertex of angle. Origin = vertex Standard Position:
More informationChapter 5. An Introduction to Trigonometric Functions 1-1
Chapter 5 An Introduction to Trigonometric Functions Copyright (c) The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1-1 5.1 A half line or all points extended from a single
More informationMATH 1113 Exam 3 Review. Fall 2017
MATH 1113 Exam 3 Review Fall 2017 Topics Covered Section 4.1: Angles and Their Measure Section 4.2: Trigonometric Functions Defined on the Unit Circle Section 4.3: Right Triangle Geometry Section 4.4:
More informationSection 14: Trigonometry Part 1
Section 14: Trigonometry Part 1 The following Mathematics Florida Standards will be covered in this section: MAFS.912.F-TF.1.1 MAFS.912.F-TF.1.2 MAFS.912.F-TF.1.3 Understand radian measure of an angle
More informationWarm Up: please factor completely
Warm Up: please factor completely 1. 2. 3. 4. 5. 6. vocabulary KEY STANDARDS ADDRESSED: MA3A2. Students will use the circle to define the trigonometric functions. a. Define and understand angles measured
More informationLesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231
1 Lesson 10.1 TRIG RATIOS AND COMPLEMENTARY ANGLES PAGE 231 What is Trigonometry? 2 It is defined as the study of triangles and the relationships between their sides and the angles between these sides.
More informationA Quick Review of Trigonometry
A Quick Review of Trigonometry As a starting point, we consider a ray with vertex located at the origin whose head is pointing in the direction of the positive real numbers. By rotating the given ray (initial
More informationMath 144 Activity #2 Right Triangle Trig and the Unit Circle
1 p 1 Right Triangle Trigonometry Math 1 Activity #2 Right Triangle Trig and the Unit Circle We use right triangles to study trigonometry. In right triangles, we have found many relationships between the
More informationSection 4.1: Introduction to Trigonometry
Section 4.1: Introduction to Trigonometry Review of Triangles Recall that the sum of all angles in any triangle is 180. Let s look at what this means for a right triangle: A right angle is an angle which
More informationIn section 8.1, we began by introducing the sine function using a circle in the coordinate plane:
Chapter 8.: Degrees and Radians, Reference Angles In section 8.1, we began by introducing the sine function using a circle in the coordinate plane: y (3,3) θ x We now return to the coordinate plane, but
More informationTrigonometric ratios provide relationships between the sides and angles of a right angle triangle. The three most commonly used ratios are:
TRIGONOMETRY TRIGONOMETRIC RATIOS If one of the angles of a triangle is 90º (a right angle), the triangle is called a right angled triangle. We indicate the 90º (right) angle by placing a box in its corner.)
More informationDAY 1 - GEOMETRY FLASHBACK
DAY 1 - GEOMETRY FLASHBACK Sine Opposite Hypotenuse Cosine Adjacent Hypotenuse sin θ = opp. hyp. cos θ = adj. hyp. tan θ = opp. adj. Tangent Opposite Adjacent a 2 + b 2 = c 2 csc θ = hyp. opp. sec θ =
More informationReview Notes for the Calculus I/Precalculus Placement Test
Review Notes for the Calculus I/Precalculus Placement Test Part 9 -. Degree and radian angle measures a. Relationship between degrees and radians degree 80 radian radian 80 degree Example Convert each
More informationName Trigonometric Functions 4.2H
TE-31 Name Trigonometric Functions 4.H Ready, Set, Go! Ready Topic: Even and odd functions The graphs of even and odd functions make it easy to identify the type of function. Even functions have a line
More informationMath 1330 Test 3 Review Sections , 5.1a, ; Know all formulas, properties, graphs, etc!
Math 1330 Test 3 Review Sections 4.1 4.3, 5.1a, 5. 5.4; Know all formulas, properties, graphs, etc! 1. Similar to a Free Response! Triangle ABC has right angle C, with AB = 9 and AC = 4. a. Draw and label
More informationThe triangle
The Unit Circle The unit circle is without a doubt the most critical topic a student must understand in trigonometry. The unit circle is the foundation on which trigonometry is based. If someone were to
More information8.6 Other Trigonometric Functions
8.6 Other Trigonometric Functions I have already discussed all the trigonometric functions and their relationship to the sine and cosine functions and the x and y coordinates on the unit circle, but let
More informationWalt Whitman High School SUMMER REVIEW PACKET. For students entering AP CALCULUS BC
Walt Whitman High School SUMMER REVIEW PACKET For students entering AP CALCULUS BC Name: 1. This packet is to be handed in to your Calculus teacher on the first day of the school year.. All work must be
More information1. The circle below is referred to as a unit circle. Why is this the circle s name?
Right Triangles and Coordinates on the Unit Circle Learning Task: 1. The circle below is referred to as a unit circle. Why is this the circle s name? Part I 2. Using a protractor, measure a 30 o angle
More informationMath 144 Activity #3 Coterminal Angles and Reference Angles
144 p 1 Math 144 Activity #3 Coterminal Angles and Reference Angles For this activity we will be referring to the unit circle. Using the unit circle below, explain how you can find the sine of any given
More informationSection 7.1. Standard position- the vertex of the ray is at the origin and the initial side lies along the positive x-axis.
1 Section 7.1 I. Definitions Angle Formed by rotating a ray about its endpoint. Initial side Starting point of the ray. Terminal side- Position of the ray after rotation. Vertex of the angle- endpoint
More informationChapter 4/5 Part 1- Trigonometry in Radians
Chapter 4/5 Part - Trigonometry in Radians Lesson Package MHF4U Chapter 4/5 Part Outline Unit Goal: By the end of this unit, you will be able to demonstrate an understanding of meaning and application
More informationMATH 181-Trigonometric Functions (10)
The Trigonometric Functions ***** I. Definitions MATH 8-Trigonometric Functions (0 A. Angle: It is generated by rotating a ray about its fixed endpoint from an initial position to a terminal position.
More informationto and go find the only place where the tangent of that
Study Guide for PART II of the Spring 14 MAT187 Final Exam. NO CALCULATORS are permitted on this part of the Final Exam. This part of the Final exam will consist of 5 multiple choice questions. You will
More informationTriangle Trigonometry
Honors Finite/Brief: Trigonometry review notes packet Triangle Trigonometry Right Triangles All triangles (including non-right triangles) Law of Sines: a b c sin A sin B sin C Law of Cosines: a b c bccos
More informationDefns An angle is in standard position if its vertex is at the origin and its initial side is on the -axis.
Math 335 Trigonometry Sec 1.1: Angles Terminology Line AB, Line segment AB or segment AB, Ray AB, Endpoint of the ray AB is A terminal side Initial and terminal sides Counterclockwise rotation results
More informationDefinitions Associated w/ Angles Notation Visualization Angle Two rays with a common endpoint ABC
Preface to Chapter 5 The following are some definitions that I think will help in the acquisition of the material in the first few chapters that we will be studying. I will not go over these in class and
More informationCW High School. Advanced Math A. 1.1 I can make connections between the algebraic equation or description for a function, its name, and its graph.
1. Functions and Math Models (10.00%) 1.1 I can make connections between the algebraic equation or description for a function, its name, and its graph. 4 Pro cient I can make connections between the algebraic
More informationTrigonometry Review Version 0.1 (September 6, 2004)
Trigonometry Review Version 0. (September, 00 Martin Jackson, University of Puget Sound The purpose of these notes is to provide a brief review of trigonometry for students who are taking calculus. The
More informationTable of Contents. Unit 5: Trigonometric Functions. Answer Key...AK-1. Introduction... v
These materials ma not be reproduced for an purpose. The reproduction of an part for an entire school or school sstem is strictl prohibited. No part of this publication ma be transmitted, stored, or recorded
More informationSecondary Math 3- Honors. 7-4 Inverse Trigonometric Functions
Secondary Math 3- Honors 7-4 Inverse Trigonometric Functions Warm Up Fill in the Unit What You Will Learn How to restrict the domain of trigonometric functions so that the inverse can be constructed. How
More informationTrigonometry, Pt 1: Angles and Their Measure. Mr. Velazquez Honors Precalculus
Trigonometry, Pt 1: Angles and Their Measure Mr. Velazquez Honors Precalculus Defining Angles An angle is formed by two rays or segments that intersect at a common endpoint. One side of the angle is called
More informationSection 6.2 Graphs of the Other Trig Functions
Section 62 Graphs of the Other Trig Functions 369 Section 62 Graphs of the Other Trig Functions In this section, we will explore the graphs of the other four trigonometric functions We ll begin with the
More informationSM 2. Date: Section: Objective: The Pythagorean Theorem: In a triangle, or
SM 2 Date: Section: Objective: The Pythagorean Theorem: In a triangle, or. It doesn t matter which leg is a and which leg is b. The hypotenuse is the side across from the right angle. To find the length
More informationLESSON 1: Trigonometry Pre-test
LESSON 1: Trigonometry Pre-test Instructions. Answer each question to the best of your ability. If there is more than one answer, put both/all answers down. Try to answer each question, but if there is
More informationTrigonometry Curriculum Guide Scranton School District Scranton, PA
Trigonometry Scranton School District Scranton, PA Trigonometry Prerequisite: Algebra II, Geometry, Algebra I Intended Audience: This course is designed for the student who has successfully completed Algebra
More information1.6 Applying Trig Functions to Angles of Rotation
wwwck1org Chapter 1 Right Triangles and an Introduction to Trigonometry 16 Applying Trig Functions to Angles of Rotation Learning Objectives Find the values of the six trigonometric functions for angles
More information4.1 Angles and Angle Measure. 1, multiply by
4.1 Angles and Angle Measure Angles can be measured in degrees or radians. Angle measures without units are considered to be in radians. Radian: One radian is the measure of the central angle subtended
More informationTrigonometric Graphs. Graphs of Sine and Cosine
Trigonometric Graphs Page 1 4 Trigonometric Graphs Graphs of Sine and Cosine In Figure 13, we showed the graphs of = sin and = cos, for angles from 0 rad to rad. In reality these graphs extend indefinitely
More informationUnit 3, Lesson 1.3 Special Angles in the Unit Circle
Unit, Lesson Special Angles in the Unit Circle Special angles exist within the unit circle For these special angles, it is possible to calculate the exact coordinates for the point where the terminal side
More information1. The Pythagorean Theorem
. The Pythagorean Theorem The Pythagorean theorem states that in any right triangle, the sum of the squares of the side lengths is the square of the hypotenuse length. c 2 = a 2 b 2 This theorem can be
More informationSummer Review for Students Entering Pre-Calculus with Trigonometry. TI-84 Plus Graphing Calculator is required for this course.
Summer Review for Students Entering Pre-Calculus with Trigonometry 1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios
More informationTrigonometry I -- Answers -- Trigonometry I Diploma Practice Exam - ANSWERS 1
Trigonometry I -- Answers -- Trigonometry I Diploma Practice Exam - ANSWERS www.puremath.com Formulas These are the formulas for Trig I you will be given on your diploma. a rθ sinθ cosθ tan θ cotθ cosθ
More informationC. HECKMAN TEST 2A SOLUTIONS 170
C HECKMN TEST SOLUTIONS 170 (1) [15 points] The angle θ is in Quadrant IV and tan θ = Find the exact values of 5 sin θ, cos θ, tan θ, cot θ, sec θ, and csc θ Solution: point that the terminal side of the
More informationTrigonometry. Secondary Mathematics 3 Page 180 Jordan School District
Trigonometry Secondary Mathematics Page 80 Jordan School District Unit Cluster (GSRT9): Area of a Triangle Cluster : Apply trigonometry to general triangles Derive the formula for the area of a triangle
More informationIn a right triangle, the sum of the squares of the equals the square of the
Math 098 Chapter 1 Section 1.1 Basic Concepts about Triangles 1) Conventions in notation for triangles - Vertices with uppercase - Opposite sides with corresponding lower case 2) Pythagorean theorem In
More informationROCKWOOD CURRICULUM WRITING PROCESS OVERVIEW
ROCKWOOD CURRICULUM WRITING PROCESS OVERVIEW Course Content Area Last Update for this Course Trigonometry Mathematics February 2009 Results of Program Evaluation Program Evaluation Recommendations Continue
More informationYou found and graphed the inverses of relations and functions. (Lesson 1-7)
You found and graphed the inverses of relations and functions. (Lesson 1-7) LEQ: How do we evaluate and graph inverse trigonometric functions & find compositions of trigonometric functions? arcsine function
More informationMath Handbook of Formulas, Processes and Tricks. Trigonometry
Math Handbook of Formulas, Processes and Tricks (www.mathguy.us) Trigonometry Prepared by: Earl L. Whitney, FSA, MAAA Version 2.1 April 10, 2017 Copyright 2012 2017, Earl Whitney, Reno NV. All Rights Reserved
More information1 Trigonometry. Copyright Cengage Learning. All rights reserved.
1 Trigonometry Copyright Cengage Learning. All rights reserved. 1.1 Radian and Degree Measure Copyright Cengage Learning. All rights reserved. Objectives Describe angles. Use radian measure. Use degree
More informationAll of my class notes can be found at
My name is Leon Hostetler. I am currently a student at Florida State University majoring in physics as well as applied and computational mathematics. Feel free to download, print, and use these class notes.
More informationMAC Module 1 Trigonometric Functions. Rev.S08
MAC 1114 Module 1 Trigonometric Functions Learning Objectives Upon completing this module, you should be able to: 1. Use basic terms associated with angles. 2. Find measures of complementary and supplementary
More informationThis unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA.
Angular Rotations This unit is built upon your knowledge and understanding of the right triangle trigonometric ratios. A memory aid that is often used was SOHCAHTOA. sin x = opposite hypotenuse cosx =
More information3.0 Trigonometry Review
3.0 Trigonometry Review In trigonometry problems, all vertices (corners or angles) of the triangle are labeled with capital letters. The right angle is usually labeled C. Sides are usually labeled with
More informationMAC Module 3 Radian Measure and Circular Functions. Rev.S08
MAC 1114 Module 3 Radian Measure and Circular Functions Learning Objectives Upon completing this module, you should be able to: 1. Convert between degrees and radians. 2. Find function values for angles
More informationAlbertson AP Calculus AB AP CALCULUS AB SUMMER PACKET DUE DATE: The beginning of class on the last class day of the first week of school.
Albertson AP Calculus AB Name AP CALCULUS AB SUMMER PACKET 2017 DUE DATE: The beginning of class on the last class day of the first week of school. This assignment is to be done at you leisure during the
More informationPre Calculus Worksheet: Fundamental Identities Day 1
Pre Calculus Worksheet: Fundamental Identities Day 1 Use the indicated strategy from your notes to simplify each expression. Each section may use the indicated strategy AND those strategies before. Strategy
More informationCopyrighted by Gabriel Tang B.Ed., B.Sc. Page 167.
lgebra Chapter 8: nalytical Trigonometry 8- Inverse Trigonometric Functions Chapter 8: nalytical Trigonometry Inverse Trigonometric Function: - use when we are given a particular trigonometric ratio and
More informationUNIT 2 RIGHT TRIANGLE TRIGONOMETRY Lesson 1: Exploring Trigonometric Ratios Instruction
Prerequisite Skills This lesson requires the use of the following skills: measuring angles with a protractor understanding how to label angles and sides in triangles converting fractions into decimals
More informationModule 4 Graphs of the Circular Functions
MAC 1114 Module 4 Graphs of the Circular Functions Learning Objectives Upon completing this module, you should be able to: 1. Recognize periodic functions. 2. Determine the amplitude and period, when given
More informationTrigonometry Summer Assignment
Name: Trigonometry Summer Assignment Due Date: The beginning of class on September 8, 017. The purpose of this assignment is to have you practice the mathematical skills necessary to be successful in Trigonometry.
More informationMA 154 Lesson 1 Delworth
DEFINITIONS: An angle is defined as the set of points determined by two rays, or half-lines, l 1 and l having the same end point O. An angle can also be considered as two finite line segments with a common
More informationCLEP Pre-Calculus. Section 1: Time 30 Minutes 50 Questions. 1. According to the tables for f(x) and g(x) below, what is the value of [f + g]( 1)?
CLEP Pre-Calculus Section : Time 0 Minutes 50 Questions For each question below, choose the best answer from the choices given. An online graphing calculator (non-cas) is allowed to be used for this section..
More informationAccel. Geometry - Concepts Similar Figures, Right Triangles, Trigonometry
Accel. Geometry - Concepts 16-19 Similar Figures, Right Triangles, Trigonometry Concept 16 Ratios and Proportions (Section 7.1) Ratio: Proportion: Cross-Products Property If a b = c, then. d Properties
More information7.1/7.2 Apply the Pythagorean Theorem and its Converse
7.1/7.2 Apply the Pythagorean Theorem and its Converse Remember what we know about a right triangle: In a right triangle, the square of the length of the is equal to the sum of the squares of the lengths
More information