1 Trigonometry -Ideas and Applications
|
|
- Shannon Rodgers
- 5 years ago
- Views:
Transcription
1 1 Trigonometry -Ideas and Applications 1.1 A second look at graphs The sine and cosine are basic entities of trigonometry, for the other four functions can be defined in terms of them. The graphs can be stretched and pulled in various ways and for many applications it is important to be able to express the corresponding functions analytically. As we have seen, the behavior of both sine and the cosine graphs on the interval [0, 2π] is repeated on all subsequent and previous intervals of length 2π. What I mean is that on each of the intervals [2π, 4π], [4π, 6π], [6π, 8π] and [ 2π, 0], [ 4π, 2π], [ 6π, 4π], the graphs look identical to the graph over the interval [0, 2π]. The restriction of the sine function to the interval [0, 2π] is defined to be the basic cycle of the sine function. Similarly the basic cycle of the cosine function is its restriction to the interval [0, 2π]. Earlier in studying general properties of functions given an arbitrary function whose graph had been altered in certain ways, we developed methods for analytically expressing a new function whose graph was the altered graph. We need to apply these ideas Stretching and shrinking vertically, reflecting about the x axis Figure 1: Graphs of y = 3 sin x in blue, y = sin x in green and y = 2 2 sin x in magenta Given either the sine function, y = sin x, or the cosine function, y = cos x, and a constant a,we know that : if a > 1, multiplication by a stretches the graph vertically if 0 < a < 1, multiplication by a shrinks the graph vertically 1
2 if a < 0, multiplication by a first reflects the graph about the x axis and then either stretches it if a < 1 or shrinks it if 1 < a < 0. Given an equation y = a sin x or y = a cos x, the number a is said to be the amplitude of the corresponding equation. Figure 2: Graph of y sin x in green and y = 2 sin x in blue Translating To translate the graph of a function vertically up, add a positive constant to the function, and to translate down, subtract a positive constant from the function definition. Thus for instance, the graph of y = sin x + 2, represents the graph of y = sin x translated two units up. Figure 3: The graph of the cosine in green, its vertical translation in magenta y = cos x+2, and its horizontal translate in blue by π 4 to the right - the graph of y = cos(x π 4 ). 2
3 To translate horizontally we have learned that to translate c units to the right, for c > 0, it is necessary to subtract c from the x variable and to translate c units to the left one adds c to the x variable. The same holds true for the sine and cosine graphs. Thus, for example, the graph of y = cos(x π 4 ) is a translation of π 4 units to the right of the graph of y = cos x. See figure Expanding and compressing horizontally, reflecting about the y axis Again from our previous discussions, we know that in general and specifically for y = sin x or y = cos x, given a constant.b Figure 4: Graphs of y = sin x in green, y = sin 2x in blue, and y = sin x 2 in magenta if b > 1, multiplication of the x variable by b compress the graph horizontally if 0 < b < 1, multiplication of the x variable by b expands the graph horizontally if b < 0, multiplication of the x variable by b first reflects the graph about the the y axis and then either compresses it or expands it according to whether b < 1 or 1 < b < 0 Given either y = sin bx or y = cos bx, the number 2π is called the period of the corresponding equation. It represents the length of the basic cycle associated with the equation. In b figure 4 we show, superimposed on the graph of y = sin x in green, the graph of y = sin 2x in blue,whose basic cycle is of length π, and the graph of y = sin x 2 in magenta, whose period is of length 4π 3
4 1.1.4 Harmonic motion Suppose you throw a stone into still pond. The ripples that gradually radiate from where the stone landed have the shape of a moving 3-dimensional sine curve. Figure 5 shows a picture of ripples in water, and figure 5 shows a mathematical model built from the sine function. Figure 5: Ripples in a pond Figure 6: Mathematical surface that models ripples - graph of z = 0.25 sin(x 2 + y 2 ) In a similar way sound creates ripples or vibrations in the air that travel radially out from the source. A tuning fork used by musicians for tuning their instruments creates vibrations in the air much like the ripples in a pond. Such physical wave phenomena are described in terms of their frequency. In these situations we consider the sine or cosine functions to be functions of time - that is, we imagine the graph of a sine or cosine function passing by us at some fixed speed - think of the movement of ripples in water. The frequency of such a curve is the number of basic cycles that pass per time unit. Usually frequency is measured 4
5 in cycles per second. When we spoke of the period of a sine or cosine curve we defined it to be the length of a basic cycle. Now if we choose to let the x axis represent time instead of physical distance and we rewrite our equations as y = a sin bt and y = a cos bt, for some constants a and b greater than zero, then the period, which we said was 2π b, now refers to the amount of time it takes a basic cycle to occur. If we measure time in seconds, then we can say that the period is the seconds required to complete one cycle. That is seconds per cycle, seconds. The reciprocal of the period then measures cycles per second, y = a sin bt and y = a cos bt this is then Example 1 : Concert pitch A frequency = b 2π cycles second cycle. Thus for the functions Most instruments in an orchestra tune to the so called concert pitch A, a sound with frequency 440 cycles per second. This means in our equations b 2π = 440, so that, b = 880π. The sine equation describing the vibration of the air is then, y = a sin 880π. The constant a is used to model the volume of the sound. Louder sounds require models with larger values of a. Example 2 : Vibrating spring Figure 7: Motion of a weight attached to a spring Imagine a weight at rest on a table but attached to a spring which is anchored in the wall somewhat to the left of the spring, see figure 7 Now pull the weight away from the wall 5
6 stretching the spring. Letting the weight go it springs back to a position to the left of its rest position and then rebounds to a position to the right of the rest position and so forth. Label the rest position zero, the compressed position to the left as -1 and the stretched position to the right +1. Once let go the weight oscillates between the -1 position and the +1 position, gradually due to friction and the nature of the spring the period of the oscillation will shorten and eventually stop. Lets assume an ideal situation in which there is no friction and no fatigue in the spring. Lets further assume that the motion has been modeled by the equation y = 7.2 sin 3πt. We then know that the amplitude a = 7.2 the period is: 2π b the frequency is 3 2 = 2π 3π = 2 3 seconds Hz ( Hz stands for cycles per second) Figure 8: Graph of y = 7.2 sin 3πt in red with a damped version in blue The graph looks like In actual fact the motion of the the weight undergoes a dampening effect whereby the distance traveled gradually decreases. In figure 8 we see the graph of y = 7.2 sin 3πt in red along with a damped version in blue. The later has been chosen to be the the graph of y = e.2(t 10) sin 3πt Inverse trigonometric functions Lets recall what we know about inverse funcitons. Given a function f : A B defined on an interval A, where B = f(a) be the set of all numbers f(x) such that x A. Then we have said that if f is injective f, there will be a function f(x) such that x A with the property that ( ) f f 1 (y) = y for every y B 6
7 ) f (f(x) 1 = x for every x A. Figure 9: Tangent on ( π 2, π 2 ) Now the problem with finding inverses of the trigonometric functions is that none of them satisfy the horizontal line test - none of them are injective. However for each of them they are injective when restricted to certain special intervals. For instance if we consider the tangent only on the interval A = ( π 2, π 2 ), then the graph look is as we see it in figure 9. The graph clearly satisfies the horizontal line test - so an inverse exists. The inverse of the tangent is denoted either as arctan or as tan 1. Since the range of the tangent is is the entire real line, in our definition of an inverse function the set B = R. We then have the relationships, arctan(tan x) = x tan(arctan y) = y Said another way, the arctan of a number y is the angle x whose tangent is y. As for the graph of the arctan, remember that in general given a function f with an inverse f 1, the graph of an f 1 is obtained by reflecting the graph of f about the diagonal line y = x. For the arctan the graph is as in figure 10. Note just as tan is asymptotic to the vertical lines x = π 2 and x = π 2 so arctan is asymptotic to the horizontal lines y = π 2 and y = π 2. The definitions other inverse trigonometric functions depends only on finding a convenient interval over which the associated trigonometric function is injective. The definiton of the inverse cotangent, written arccotan or cot 1, relies on restricting the definition of the cotangent to the interval (0, π). The arccotan then has domain the whole real line and range the interval (0, π). See figures 11 and 12. 7
8 Figure 10: Graph of arctan. The definition of the inverse sine, written arcsin or sin 1, relies on restricting the sine function to the interval [ π 2, π 2 ]. Since the sine has range [ 1, 1], the arcsine then has domain [ 1, 1] and range [ π 2, π 2 ]. Figure 13 shows the graph of the sine in blue and the graph of the arcsine in purple. Note that the sine is injective on the interval [ π 2, π 2 ]. For the inverse cosine, written arc-cosine or cos 1, the domain of the cosine is restricted to the interval [0, π]. The arc-cosine then has this as its range. See figures 14 and 15. For the inverse secant, written arcsec or sec 1, the chosen domain for the secant consists of the intervals [0, π 2 ) and on ( π 2, π] on which the secant is injective. The inverse secant then has domain the range of the secant, namely the two intervals [0, π 2 ) and ( π 2, π], and the range of the inverse secant is the domain of the secant namely the intervals[0, π 2 ) and ( π 2, π]. See figures 16 and 17. The arc-cosecant function may be similarly defined by considering the cosecant function restricted to the intervals [ π 2, 0) and (0, π 2 ). Example 3 Write each of: (a) cos(arcsin 4x) and (b) tan(arcsin 4x) as a function not involving a trigonometric function. What is the domain of the function? Solution: Begin by setting θ equal to the angle whose sine is 4x. In other words set θ = arcsin 4x. Next draw a right triangle as in figure 18. For convenience the triangle is chosen so that the hypotenuse has length one. If this is true and of sin θ = 4x, the 8
9 Figure 11: Cotangent is injective on (0, π) 9
10 Figure 12: Arccotan is defined on all of R with range (0, π) Figure 13: Sine in blue and arcsine in purple; sine is injective on [ π 2, π 2 ], arcsine defined on [ 1, 1] with range [ π 2, π 2 ] 10
11 Figure 14: Cosine injective on[0, π] Figure 15: Arc-cosine with domain[ 1, 1] and range [0, π] 11
12 Figure 16: Secant defined on [0, π 2 ) and on ( π 2, π]. Range:[1, ) (, 1] Figure 17: Arcsecant defined on [1, ) (, 1] with range [0, π 2 ) and ( π 2, π]. 12
13 Figure 18: Right triangle with θ = arcsin 4x. opposite side must have length 4x. Using the Pythagorean theorem, the adjacent side is calculated to have length 1 (4x) 2. We can now read off the answers. First, in order that 1 (4x) 2 not be negative, we must have (4x) 2 < 1 or x 2 < 1 16, which implies that 1 4 x 1 4. Then cos(arcsin 4x) = 1 (4x) 2 tan(arcsin 4x) = 4x 1 (4x) 2 Example 4 A sailor sailing single-handed about the globe encountered a ferocious storm in the south Atlantic. All his electronic equipment was damaged with the exception of a scientific calculator, his compass and his watch which had been set to Greenwich mean time. With the help of his compass he was able to determine with reasonable precision the time ( that is the local Noon) when the sun was directly south. To determine his latitude he however need to determine the elevation of the sun at Noon. He was able to determine that a vertical pole 2 meters high cast a shadow at Noon that was precisely 1.8 meters long. With this information he was able to calculate the elevation of the sun. How was this done? See figure 19. Solution: The sailor being schooled in trigonometry realized that if θ is the angle of elevation, it must be true that the tangent of θ must be = 10 9 = Taking out his calculator he entered and then pushed the inverse tangent button to find the angle whose tangent is The result was tan 1 ( ) = o. 13
14 Figure 19: Find the angle θ 1.2 Further Applications Figure 20: y = sin( x)is the equation of the red curve - when moved π to the left it has equation y = sin(π θ and corresponds to the blue curve y = sin θ Trigonometry allows an easy way to calculate the area of any triangle. First we need to observe that starting with the graph of the sine y = sin θ, reflecting about the x axis gives the graph of y = sin θ which we also know from the section on identities is the same as the graph of y = sin( θ). Using this equation, if we were to translate the graph π units to the left, the corresponding equation is then y = sin( θ + π). The graph we see is precisely the graph of sine. That is we got back to where we had started. See figure 20. Thus we can say that sin θ = sin(π θ). 14
15 1.2.1 Area of a triangle Given an arbitrary triangle we have learned that its area can be calculated by the formula A = base height 2. The difficulty with this formula that except for the case of right triangles, the height of the triangle does not correspond to the length of one of the edges. And if the lengths of edges is all the information available, one is stuck - that is, if one does not know a bit of trigonometry. Figure 21: A base angle θ may be acute or obtuse - in which case the complimentary angle π θ is acute Given a triangle lets let θ be one of the base angles. As we see in figure 21 the angle may be acute (less than 90 o ) or obtuse (greater than 90 o ). In the first case we know that sin θ = h a and in the second case we know that sin(π θ) = h a. But as we have seen, sin(π θ) = sin θ so thus, in both cases, we conclude that h = a sin θ. We then have in either case that the area is 1 2ab sin θ. This is expressed more formally as follows Result 5 Area of triangle The area A of a triangle with two sides of length a and b and an included angle θ is A = 1 ab sin θ Law of sines The law of sines says that for a given triangle ABC the ratio of the sine of angle to the length of its opposite side is independent of the chosen angle. In other words, if a, b and c are the lengths of the sides opposite the angle A, B and C, See figure 22. sin A a = sin B b = sin C. c The law of sines is used in situations where given a triangle it is possible to calculate the ratio of the sine of one of the angles to the length of its opposite side. Then as long as one more piece of information is present, the length of another side or the measurement 15
16 Figure 22: Triangle with angles A, B, and C with opposite sides of length a,b, and c of another angle, all the other dimensions of the triangle can be calculated. For instance suppose we know the size of angle A and the length a of its opposite side. Then if we know the length b of one of the other sides, we have sin A a = sin B b so that sin B = b sin A a. Thus the angle at B has measurement arcsin(b sin A a ). Knowing B we then know the measurement of the angle at C and hence sin C and ultimately the length c of the side opposite C. Here are some examples. Example 6 Given a triangle ABC suppose that A = 83 o, B = 55 o and b = 18. Find the lengths of the other sides. Solution: Since C = 180 ( ) = 42 o, we have sin 55 o 18 = sin 83o a and sin 55 o 18 = sin 42o c Thus a = 18 sin 83o sin 55 o 21.8 and c = 18 sin 42o sin 55 o Example 7 A construction company was awarded a contract by the provincial government to build a foot bridge across a gully on the west shore of Vancouver Island. Although the company knew that the bridge was to go from point A on one side of the gully to point B on the other, they did not know how long the bridge should be. In order to determine the length they set a marker C on the same side as the point B and measured the distance between A and B to be 10 meters. From the other side, with surveying equipment they were able to measure the angel at A, BAC, to be 12 o. The also measured the angle at C, BCA, to be 42 o. With this information they were able to find what the length of the bridge should be. What is this length? Solution: c sin 42 = 10 o sin 12 o We know from the law of sines that if c is the length of the bridge, then 10 sin 42o so that c = sin meters. See figure 23. o 16
17 Figure 23: Calculate the length of the bridge from A to B Using our previous result for the area of a general triangle, the law of sines may be easily proved Proof: Considering figure 22 we can calculate the area in three different ways, depending on which angle we wish to consider. Multiplying each term by 2 abc Area = 1 2 bc sin A = 1 2 ac sin B = 1 ab sin C. 2 gives the result Law of Cosines The law of cosines may be considered a generalization of the Pythagorean theorem. Given an arbitrary triangle as in figure 22, if the angle at C were a right angle then by the Pythagorean theorem c 2 = a 2 + b 2. But what if the angle at C is not a right angle? The law of cosines tells us that c 2 = a 2 + b 2 2ab cos C and in the case that C = 90 o we have cos C = 0 and the law of cosines reduces to the Pythagorean theorem. 17
18 Figure 24: Two boats one traveling N40 o E the other travelings60 o E. Figure 25: position of boats after one hour 18
19 The law of cosines also lets tells us that the converse of the Pythagorean theorem is true, for we then know that if C is not a right angle, then cos C 0 and c 2 a 2 + b 2. From our work in logic we know that the contrapositive of this last statement must also be true - namely, if c 2 = a 2 + b 2, then C is a right angle. This last statement is the converse of the Pythagorean theorem. Here is an example to show how the law of cosines may be applied. Example 8 Two boats leave the same harbor at Noon. Boat A travels at speed of 50 km/h in the direction N45 o E and the other, boat B, travels at a speed of 35 km/h in a direction S60 o E. What is the distance between them at 1:00 P.M.? Solution: We see that the angle between the paths of the two boats is 45 o +30 o = 75 o, and after 1 hour boat A has traveled 50 km and boat B has traveled 35 km. Letting C stand for their common starting point we have a diagram as in figure 25. By the law of cosines the distance c between the two boats is c = (50)(35) cos 75 o 53.1km. Example 9 Given a regular pentagon whose 5 sides are each of length 1. Let A, B, and C be three vertices as in figure 26 and let D be the mid-point of the line segment BC. What is the length b of the line segment AD? Solution: Using the fact that the angle at each of the vertices of a regular pentagon is 108 o, a fact that is not difficult to prove using elementary geometry, by the law of cosines b 2 = (1)(1 ) cos Thus b = 1.25 cos Figure 26: For a regular pentagon with sides of length 1, find length b of the segment AD 19
Section 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 informationGraphing Trigonometric Functions: Day 1
Graphing Trigonometric Functions: Day 1 Pre-Calculus 1. Graph the six parent trigonometric functions.. Apply scale changes to the six parent trigonometric functions. Complete the worksheet Exploration:
More informationPRECALCULUS MATH Trigonometry 9-12
1. Find angle measurements in degrees and radians based on the unit circle. 1. Students understand the notion of angle and how to measure it, both in degrees and radians. They can convert between degrees
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 informationBasic Graphs of the Sine and Cosine Functions
Chapter 4: Graphs of the Circular Functions 1 TRIG-Fall 2011-Jordan Trigonometry, 9 th edition, Lial/Hornsby/Schneider, Pearson, 2009 Section 4.1 Graphs of the Sine and Cosine Functions Basic Graphs of
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 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 informationChapter Nine Notes SN P U1C9
Chapter Nine Notes SN P UC9 Name Period Section 9.: Applications Involving Right Triangles To evaluate trigonometric functions with a calculator, there are a few important things to know: On your calculator,
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 informationTRIGONOMETRY. Meaning. Dear Reader
TRIGONOMETRY Dear Reader In your previous classes you have read about triangles and trigonometric ratios. A triangle is a polygon formed by joining least number of points i.e., three non-collinear points.
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 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 informationAlgebra 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 informationThis is called the horizontal displacement of also known as the phase shift.
sin (x) GRAPHS OF TRIGONOMETRIC FUNCTIONS Definitions A function f is said to be periodic if there is a positive number p such that f(x + p) = f(x) for all values of x. The smallest positive number p for
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 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 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 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 informationMathematical Techniques Chapter 10
PART FOUR Formulas FM 5-33 Mathematical Techniques Chapter 10 GEOMETRIC FUNCTIONS The result of any operation performed by terrain analysts will only be as accurate as the measurements used. An interpretation
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 informationRight Triangle Trigonometry
Right Triangle Trigonometry 1 The six trigonometric functions of a right triangle, with an acute angle, are defined by ratios of two sides of the triangle. hyp opp The sides of the right triangle are:
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 informationAlgebra II. Slide 1 / 162. Slide 2 / 162. Slide 3 / 162. Trigonometric Functions. Trig Functions
Slide 1 / 162 Algebra II Slide 2 / 162 Trigonometric Functions 2015-12-17 www.njctl.org Trig Functions click on the topic to go to that section Slide 3 / 162 Radians & Degrees & Co-terminal angles Arc
More informationName: Block: What I can do for this unit:
Unit 8: Trigonometry Student Tracking Sheet Math 10 Common Name: Block: What I can do for this unit: After Practice After Review How I Did 8-1 I can use and understand triangle similarity and the Pythagorean
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 informationIntro Right Triangle Trig
Ch. Y Intro Right Triangle Trig In our work with similar polygons, we learned that, by definition, the angles of similar polygons were congruent and their sides were in proportion - which means their ratios
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 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 information9.1 Use Trigonometry with Right Triangles
9.1 Use Trigonometry with Right Triangles Use the Pythagorean Theorem to find missing lengths in right triangles. Find trigonometric ratios using right triangles. Use trigonometric ratios to find angle
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 informationChapter 4: Triangle and Trigonometry
Chapter 4: Triangle and Trigonometry Paper 1 & 2B 3.1.3 Triangles 3.1.3 Triangles 2A Understand a proof of Pythagoras Theorem. Understand the converse of Pythagoras Theorem. Use Pythagoras Trigonometry
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Convert the angle to decimal degrees and round to the nearest hundredth of a degree. 1)
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 informationCK-12 Geometry: Inverse Trigonometric Ratios
CK-12 Geometry: Inverse Trigonometric Ratios Learning Objectives Use the inverse trigonometric ratios to find an angle in a right triangle. Solve a right triangle. Apply inverse trigonometric ratios to
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 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 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 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 informationChapter 4. Trigonometric Functions. 4.6 Graphs of Other. Copyright 2014, 2010, 2007 Pearson Education, Inc.
Chapter 4 Trigonometric Functions 4.6 Graphs of Other Trigonometric Functions Copyright 2014, 2010, 2007 Pearson Education, Inc. 1 Objectives: Understand the graph of y = tan x. Graph variations of y =
More informationarchitecture, physics... you name it, they probably use it.
The Cosine Ratio Cosine Ratio, Secant Ratio, and Inverse Cosine.4 Learning Goals In this lesson, you will: Use the cosine ratio in a right triangle to solve for unknown side lengths. Use the secant ratio
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 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 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 information10-1. Three Trigonometric Functions. Vocabulary. Lesson
Chapter 10 Lesson 10-1 Three Trigonometric Functions BIG IDEA The sine, cosine, and tangent of an acute angle are each a ratio of particular sides of a right triangle with that acute angle. Vocabulary
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 informationSolving Trigonometric Equations
OpenStax-CNX module: m49398 1 Solving Trigonometric Equations OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section, you
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 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 informationUnit 3 Trig II. 3.1 Trig and Periodic Functions
Unit 3 Trig II AFM Mrs. Valentine Obj.: I will be able to use a unit circle to find values of sine, cosine, and tangent. I will be able to find the domain and range of sine and cosine. I will understand
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 informationName: Teacher: Pd: Algebra 2/Trig: Trigonometric Graphs (SHORT VERSION)
Algebra 2/Trig: Trigonometric Graphs (SHORT VERSION) In this unit, we will Learn the properties of sine and cosine curves: amplitude, frequency, period, and midline. Determine what the parameters a, b,
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 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 Ratios. For each of the right triangles below, the labelled angle is equal to 40. Why then are these triangles similar to each other?
Name: Trigonometry Ratios A) An Activity with Similar Triangles Date: For each of the right triangles below, the labelled angle is equal to 40. Why then are these triangles similar to each other? Page
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 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 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 informationChapter 15 Right Triangle Trigonometry
Chapter 15 Right Triangle Trigonometry Sec. 1 Right Triangle Trigonometry The most difficult part of Trigonometry is spelling it. Once we get by that, the rest is a piece of cake. efore we start naming
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 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 information1. GRAPHS OF THE SINE AND COSINE FUNCTIONS
GRAPHS OF THE CIRCULAR FUNCTIONS 1. GRAPHS OF THE SINE AND COSINE FUNCTIONS PERIODIC FUNCTION A period function is a function f such that f ( x) f ( x np) for every real numer x in the domain of f every
More informationSolving Right Triangles. How do you solve right triangles?
Solving Right Triangles How do you solve right triangles? The Trigonometric Functions we will be looking at SINE COSINE TANGENT The Trigonometric Functions SINE COSINE TANGENT SINE Pronounced sign TANGENT
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 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. 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 informationG.8 Right Triangles STUDY GUIDE
G.8 Right Triangles STUDY GUIDE Name Date Block Chapter 7 Right Triangles Review and Study Guide Things to Know (use your notes, homework, quizzes, textbook as well as flashcards at quizlet.com (http://quizlet.com/4216735/geometry-chapter-7-right-triangles-flashcardsflash-cards/)).
More informationAW Math 10 UNIT 7 RIGHT ANGLE TRIANGLES
AW Math 10 UNIT 7 RIGHT ANGLE TRIANGLES Assignment Title Work to complete Complete 1 Triangles Labelling Triangles 2 Pythagorean Theorem 3 More Pythagorean Theorem Eploring Pythagorean Theorem Using Pythagorean
More informationIntro Right Triangle Trig
Ch. Y Intro Right Triangle Trig In our work with similar polygons, we learned that, by definition, the angles of similar polygons were congruent and their sides were in proportion - which means their ratios
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 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 informationAWM 11 UNIT 4 TRIGONOMETRY OF RIGHT TRIANGLES
AWM 11 UNIT 4 TRIGONOMETRY OF RIGHT TRIANGLES Assignment Title Work to complete Complete 1 Triangles Labelling Triangles 2 Pythagorean Theorem Exploring Pythagorean Theorem 3 More Pythagorean Theorem Using
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 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 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 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 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 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 informationLesson 26 - Review of Right Triangle Trigonometry
Lesson 26 - Review of Right Triangle Trigonometry PreCalculus Santowski PreCalculus - Santowski 1 (A) Review of Right Triangle Trig Trigonometry is the study and solution of Triangles. Solving a triangle
More informationUNIT 5 TRIGONOMETRY Lesson 5.4: Calculating Sine, Cosine, and Tangent. Instruction. Guided Practice 5.4. Example 1
Lesson : Calculating Sine, Cosine, and Tangent Guided Practice Example 1 Leo is building a concrete pathway 150 feet long across a rectangular courtyard, as shown in the following figure. What is the length
More informationSecondary Mathematics 3 Table of Contents
Secondary Mathematics Table of Contents Trigonometry Unit Cluster 1: Apply trigonometry to general triangles (G.SRT.9)...4 (G.SRT.10 and G.SRT.11)...7 Unit Cluster : Extending the domain of trigonometric
More information2.3 Circular Functions of Real Numbers
www.ck12.org Chapter 2. Graphing Trigonometric Functions 2.3 Circular Functions of Real Numbers Learning Objectives Graph the six trigonometric ratios as functions on the Cartesian plane. Identify the
More informationYou ll use the six trigonometric functions of an angle to do this. In some cases, you will be able to use properties of the = 46
Math 1330 Section 6.2 Section 7.1: Right-Triangle Applications In this section, we ll solve right triangles. In some problems you will be asked to find one or two specific pieces of information, but often
More informationYoungstown State University Trigonometry Final Exam Review (Math 1511)
Youngstown State University Trigonometry Final Exam Review (Math 1511) 1. Convert each angle measure to decimal degree form. (Round your answers to thousandths place). a) 75 54 30" b) 145 18". Convert
More information5B.4 ~ Calculating Sine, Cosine, Tangent, Cosecant, Secant and Cotangent WB: Pgs :1-10 Pgs : 1-7
SECONDARY 2 HONORS ~ UNIT 5B (Similarity, Right Triangle Trigonometry, and Proof) Assignments from your Student Workbook are labeled WB Those from your hardbound Student Resource Book are labeled RB. Do
More informationUnit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8)
Unit O Student Success Sheet (SSS) Right Triangle Trigonometry (sections 4.3, 4.8) Standards: Geom 19.0, Geom 20.0, Trig 7.0, Trig 8.0, Trig 12.0 Segerstrom High School -- Math Analysis Honors Name: Period:
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 informationUnit 4 Graphs of Trigonometric Functions - Classwork
Unit Graphs of Trigonometric Functions - Classwork For each of the angles below, calculate the values of sin x and cos x (2 decimal places) on the chart and graph the points on the graph below. x 0 o 30
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 informationTImath.com Algebra 2. Proof of Identity
TImath.com Algebra Proof of Identity ID: 9846 Time required 45 minutes Activity Overview Students use graphs to verify the reciprocal identities. They then use the handheld s manual graph manipulation
More informationMath 1330 Final Exam Review Covers all material covered in class this semester.
Math 1330 Final Exam Review Covers all material covered in class this semester. 1. Give an equation that could represent each graph. A. Recall: For other types of polynomials: End Behavior An even-degree
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 information2.0 Trigonometry Review Date: Pythagorean Theorem: where c is always the.
2.0 Trigonometry Review Date: Key Ideas: The three angles in a triangle sum to. Pythagorean Theorem: where c is always the. In trigonometry problems, all vertices (corners or angles) of the triangle are
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 informationabout touching on a topic and then veering off to talk about something completely unrelated.
The Tangent Ratio Tangent Ratio, Cotangent Ratio, and Inverse Tangent 8.2 Learning Goals In this lesson, you will: Use the tangent ratio in a right triangle to solve for unknown side lengths. Use the cotangent
More informationA-C Valley Junior-Senior High School
Course of Study A-C Valley Junior-Senior High School Page 1 of 12 Applied Trigonometry (NAME OF COURSE) GRADE LEVEL(S): 11 Educational Curriculum Level Person(s) Revising Curriculum (List Names) 1. Junior-Senior
More information5.5 Multiple-Angle and Product-to-Sum Formulas
Section 5.5 Multiple-Angle and Product-to-Sum Formulas 87 5.5 Multiple-Angle and Product-to-Sum Formulas Multiple-Angle Formulas In this section, you will study four additional categories of trigonometric
More informationPart Five: Trigonometry Review. Trigonometry Review
T.5 Trigonometry Review Many of the basic applications of physics, both to mechanical systems and to the properties of the human body, require a thorough knowledge of the basic properties of right triangles,
More information1. Be sure to complete the exploration before working on the rest of this worksheet.
PreCalculus Worksheet 4.1 1. Be sure to complete the exploration before working on the rest of this worksheet.. The following angles are given to you in radian measure. Without converting to degrees, draw
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 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 information