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, 98. Pg. 265: Vocab: 3-4. Ex: 23-28, 30-32, 61-67, 102, 114, 118. 2/14 Thursday 4.1 & 4.4 2/19 Tuesday 4.1 & 4.2 Pg. 294: Vocab: 1-7. Ex: 1, 4, 7, 10, 13-16, 18, 20, 37-40, 46, 48, 53-55, 65, 66, 72, 77-83, 111. Pg. 265: Vocab: 4-8. Ex: 3-9, 11, 12, 18, 29, 34, 39-41, 43-45, 47, 48, 56, 57, 72, 74, 76, 82, 84, 90, 92. Pg. 274: Vocab: 1-3. Ex: 18, 20, 24, 29, 30, 54, 56, 58, 90. 4.3 2/21 Thursday Review Review Worksheet Extra Credit: Pg. 344: 4-36 by 4 s, 42-70 by 4 s, 82, 88, 94, 100. 2/25 Monday Test Test 4.1 & 4.4 4.1 & 4.2 Review Notes Extra Credit Notes: 1 P a g e
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Chapter 4: Trigonometric Functions 4.3 Right Triangle Trigonometry Basic Right Triangle Trigonometry The basic right triangle trigonometric ratios are given by SOH-CAH-TOA For reciprocal functions, you must turn the primary trig ratios upside down. So, Example 1: Use the right triangle below to find the exact values (trig ratios) of the six trigonometric functions of. There are two types of special triangles for which you should memorize side lengths to help you build trig ratios. These are 45-45 -90 and 30-60 -90 triangles. Example 2: Fill in the sides of the triangles below. Then find the trig ratios for each acute angle. 3 P a g e
It is possible to find exact values of trig ratios when using special triangles or right triangles with two given sides. However, you may need to find approximate values of trig ratios to solve real-world problems. Your calculator will find approximate trig ratios in decimal form. Make sure your calculator is set in the correct mode (degree or radian). Example 3: Find the following trig ratios using your calculator. a. b. c. d. (Remember that e. f. A major application of trigonometry is solving for some (or all) of the missing parts of a right triangle. This process is called solving right triangles. In addition to using trigonometric ratios, you will have to use some geometry to solve right triangles. On common method is the Pythagorean Theorem. Example 4: Find the missing part labeled for each triangle below. 4 P a g e
Example 5: Find the values of the six trigonometric functions of θ in the right triangle shown. sin θ = csc θ = θ 15 cos θ = sec θ = tan θ = cot θ = 9 Example 6: Sketch a right triangle corresponding to the trigonometric function of the acute angle θ. Use the Pythagorean Theorem to determine the third side and then find the other 5 trigonometric functions of θ. a. tan θ = 3 Example 7: A 30-meter line is used to tether a helium filled balloon. Because of a breeze, the line makes an angle of approximately 75 with the ground. a. Draw a right triangle that gives a visual representation of the problem. Label the known parts and use a variable for the unknown parts. b. What is the height of the balloon? (Use a trigonometric function to write an equation involving the unknown height.) 5 P a g e
Angle of Elevation: The angle measured from a horizontal upward toward an object. (angle going up) Angle of Depression: The angle measured from a horizontal downward toward an object. (angle going down) Example 8: A surveyor standing 100 feet from the base of a large tree measures the angle of elevation to the top of the tree to be 53.5. How tall is the tree? Example 9: A sonar operator on a ship detects a submarine at a distance of 500 meters and an angle of depression of 40. How deep is the submarine? 6 P a g e
4.1 & 4.4 Degree Measure and Trigonometric Functions of any Angle Angles An angle is formed by rotating a ray about its endpoint. The starting point of the ray is the side. The final position of the ray is the side. The point of rotation is called the of the angle. The amount and direction of rotation is the of the angle. A (1) degree is of a complete rotation. If the rotation is counter-clockwise (CCW), the measure of the angle is. If the rotation is clockwise (CW), the measure of the angle is. Angles are named with Greek letters (α, β, θ) or Capital letters (A, B, C). An angle in standard position is said to lie in the quadrant that contains its terminal side An angle in standard position where its terminal side lies on the x or y- axis is called a quadrantal angle Angles between 0 and 90 ( ) are acute Angles between 90 and 180 ( )are obtuse With calculators it is convenient to use decimal degrees to denote fractional parts of degrees. Historically, however, fractional parts of degrees were expressed in minutes and seconds ( for minutes, for seconds). 1 = one minute = ( 1 = one second = ( 64 degrees, 32 minutes and 47 seconds can be written as 64 32 47 Example 1: a. Convert 64 32 47 to decimal degree form By hand: b. Convert 152 15 29 to decimal degree form By Calculator: By Calculator: (2 nd APPS for angle and min.)(alpha + for sec.) Example 2: Convert 30.023 to DMS (In the angle menu) 7 P a g e
Reference Angles Let θ be an angle in standard position. Its reference angle is the acute angle θ formed by the terminal side of θ and the horizontal (x) axis. θ is always positive. Example 3: Sketch angles in standard position having the following measures, and find their reference angles. a. 315 b. -150 c. 420 A reference angle can be used to form a reference triangle whose trig ratios are the same as those for the actual angle. A reference triangle is formed by drawing a vertical segment from the terminal side of the angle to the x-axis, forming a right triangle. For a reference triangle, the hypotenuse is always considered positive. A leg is considered to be positive if it is to the right of or above the origin. It is considered negative if it is to the left or below the origin. Example 4: Use the reference angles from Example 3 to build reference triangles, then find the following trig ratios. Be careful with positive/negative signs! a. sin 315 b. sin (-150 ) c. sin 420 cos 315 cos (-150 ) cos 420 tan 315 tan (-150 ) tan 420 Example 5: Let (-3, 4) be a point on the terminal side of θ. Find the sine, cosine, tangent, cosecant, secant and cotangent of θ. First draw θ with terminal side on (-3, 4). Example 6: Find the values of all six trig functions of if and. 8 P a g e
If an angle does not have a reference angle of 30, 45, or 60 you can still find its trig ratios (in decimal form) by using a calculator. Example 7: Use a calculator to find the following: The signs of trigonometric function values in the four quadrants can be determined easily from the definitions of the functions. For instance, because, cos θ is positive wherever x > 0, which is in quadrant I and IV. (Remember, r is always positive.) S A T C If angles in standard position share the same terminal side, they are called coterminal angles. The angles in the diagram below have measures of 30, 390, and -330. They are coterminal. Coterminal angles have measures which differ by multiples of 360. Thus, you can find angles which are coterminal to a given angle by adding or subtracting 360 as many times as you wish. Finding Reference Angles: Example 8: Draw θ and then find the reference angle θ for each of the following. Also, find one negative and one positive coterminal angle. a. θ = 300 b. θ = c. θ = 405 9 P a g e
4.1 & 4.2 Quadrant Angles and Radian Measure for Angles Suppose an angle in standard position has its reference triangle labeled with x, y and r as shown in the figure. Then using SOH-CAH-TOA: and Now, suppose you draw a circle with radius 1, so that it circumscribes the reference triangle as shown. This circle is called a unit circle, and it is useful in developing the trig ratios for the quadrantal angles ( -270, -180, -90, 0, 90, 180, 270, ). In the unit circle, r=1 so and Example 1: Use the unit circle shown to do the following: a. Plot points at every intersection of the unit circle with an axis. b. Label these points using ordered pairs. c. Fill in the following table. (Remember for the unit circle, and ) 1-1 1-90 0 90 180 270 360 sin cos -1 tan Quadrant angles do not form reference angles, so it is important to think of a unit circle when finding their trig ratios without a calculator. For any other special angles (multiples of 30, 45 or 60 you should build reference triangles. Example 2: Find the following trig ratios: a. tan 720 b. csc (-270 ) c. sec 450 d. cos 540 e. cot (-90 ) f. sin 210 10 P a g e
In Geometry, you probably always measured angles in degrees. Many applications in trigonometry use angles measured in degrees. However, upper level mathematics courses almost always use angles which are measure in radians. One radian is defined to be the measure of the central angle (of a circle) that intercepts an arc s which is equal in length to the radius of the circle. Since the circumference of a circle is 2 r, it takes 2 arc lengths of s=r to go all the way around the circle. Thus, 360 =2 radians. One radian = 57.2958. Dividing both sides of both sides of 360 = 2 by 2, you get a very convenient equation for converting angle measurement types: Why Radian's not degrees Online Radian Explanation Since a circle s circumference = 2πr, a full circle s angle measure is 2π radians. (Just over 6 radius lengths, 6.283 radians) Conversion between Degrees and Radians 1. To convert Degrees to Radians Multiply degrees by 2. To convert Radians to Degrees Multiply radians by WHY are these the conversion equations? Example 3: Convert the following, and tell which quadrant it lies in. a. 135 b. -270 c. d. 2 rads Angles between 0 and ( ) are acute Angles between and π ( )are obtuse Note: when no units of angle measure are specified, radian measure is implied. Some common angles measured in radians: 11 P a g e
Some angles occur so often in trigonometry that you should try to memorize the degree-radian equivalencies. They are fairly easy to develop if you do forget them. Example 4: Find the following degree-radian equivalencies. Remember that 180 = radians. a. 90 = b. 30 = c. 60 = d. 45 = Example 5: Sketch each angle in standard position, and find the trig ratios asked for. a. b. sin tan = cos tan = When finding coterminal angles for angles expressed in radian measure, you can add or subtract 2 as many times as you need to. Example 6: Sketch each angle in standard position, and find one positive and one negative coterminal angle for each. (Express your answers in radian measure) a. b. Your calculator will easily find trig ratios for any angle measured in either degrees or radians. However, you must let the calculator know which type of measure that you are using. The two measures (radian and degree) are both found in the calculator s MODE. Example 7: Use your calculator to find the following to 3 decimal place accuracy. Note: When degrees are not specifically indicated, angle measures are considered to be in radians. a. cos 3.725 b. tan( ) c. csc 2.621 Arc Length For a circle of radius r, a central angle θ intercepts an arc of length s given by s=r θ Where θ is measured in radians. If r = 1 then s = θ, and the radian measure of θ equals the arc length. Example 8: If a clock hand moves from 12:00 noon to 12:20 pm and the minute hand is 6 inches long, what is the length of the arc it sweeps out? 12 P a g e
Example 9: A circle has a radius of 4 inches. Find the length of the arc intercepted by a central angle of 240. Example 10: Glasgow, MT is due north of Albuquerque, NM. Find the distance between Glasgow (48 9 ) and Albuquerque (35 5 ). Assume the radius of the earth is 3960 miles. Even and Odd Trig Functions The cosine and secant functions are even. cos(-t) = cos t sec (-t) = sec t The sine, cosecant, tangent and cotangent functions are odd. sin(-t) = - sin t csc (-t) = - csc t tan(-t) = - tan t cot (-t) = - cot t Example 11: Given cos t = -3/4 find the following: a. ( b. ( Example 12: Evaluate the following a. ( b. ( c. ( d. ( 13 P a g e