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Section 5.2: Trigonometric Functions of Angles Objectives: Upon completion of this lesson, you will be able to: find the values of the six trigonometric functions for an angle with given conditions given an angle and the length of one side of a right triangle, find the length of any missing side solve right-triangle application problems know the exact values of the six trigonometric functions for 30, 45, 60 and the quadrantal angles (angles also in radian form) find the trigonometric function of any given angle in radians or degrees find the quadrant containing an angle with given parameters use the reciprocal, tangent and cotangent, and Pythagorean identities to simplify expressions, verify identities, or write trigonometric expressions in terms of a different trigonometric function. Required Reading, Video, Tutorial Read Swokowski/Cole: Section 5.2 Watch the Trig Introduction Video Watch the Special Angles Video Watch the Section 5.2 Video Complete the Section 5.2 Before Class WA assignment Discussion Section 5.2 is packed with material, and this material is fundamental to understanding any future trigonometric concepts. Please read the section thoroughly and pay close attention to all 12 examples in the text. Analytic geometry; which is the use of algebra to study geometric properties and operations on symbols defined in a coordinate system; is fundamental to the study of trigonometry. Following is a right triangle on a coordinate grid in a circle of radius r. The circle and the triangle have the point P x, y in common. Also pictured is the same right triangle without the grid or circle. P(x, y) hypotenuse r y x adjacent side opposite side Math130: Section 5.2 – Trigonometric Functions of Angles The six trigonometric functions - sine, cosine, tangent, cotangent, secant, and cosecant - are defined as ratios of the lengths of the sides of a right triangle. c = 17 Trigonometric Ratios/Functions a=8 b Using x, y notation from the figure above, the six trigonometric ratios or functions are defined as follows. sin y opposite r hypotenuse csc 1 r sin y cos x adjacent r hypotenuse sec 1 r cos x tan y sin opposite x cos adjacent cot 1 cos x tan sin y Note the relationships between the trigonometric ratios. cosecant is the reciprocal of sine secant is the reciprocal of cosine tangent is the ratio of sine divided by cosine cotangent is the reciprocal of tangent, which is also cosine divided by sine Notation: The common labeling of right triangles is to identify the right angle as angle (gamma) with the two acute angles labeled (alpha) and (beta). The hypotenuse, the side opposite the right angle , shall be labeled c. The side opposite angle shall be labeled a, and the side opposite angle shall be labeled b. c b a Example 1: For the given right triangle, find the 6 trig functions for angle and angle . Solution: Before setting up the ratios, it is necessary to find the length of side b. This is accomplished by using the Pythagorean Theorem. a2 b2 c2 82 b 2 17 2 b 2 225 b 15 b 15 Choose +15 since b represents a length. ©2014;Dr. B. Shryock Page 2 of 7 Math130: Section 5.2 – Trigonometric Functions of Angles For angle opp 8 sin hyp 17 adj 15 cos hyp 17 opp 8 tan adj 15 For angle opp 15 sin hyp 17 adj 8 cos hyp 17 opp 15 tan adj 8 17 8 17 sec 15 15 cot 8 csc 17 15 17 sec 8 8 cot 15 csc Notice that the definition of the trigonometric function remains constant regardless of the angle used; however, the opposite and adjacent sides are redefined relative to the chosen angle. Also notice that sin cos and that sin cos . This is due to the fact that and are complementary angles and we are dealing with cofunctions. The topic of cofunction will be covered in Section 6.3. It is helpful to set up the six trigonometric functions in table form as in the above example for easy computation. An expedient method to use when working with many problems in trigonometry is to draw a picture of the situation. It is critical to draw the picture correctly, especially when locating angles in a particular quadrant to construct a triangle. The text summarizes the signs of the trigonometric functions on page 344 and in Figure 16 on page 345. It helps to remember that the cosine function is directly related to the x coordinate and will therefore have the same sign as the x coordinate has when plotting points in the four quadrants. The same holds true for the sine function, it is directly related to the y coordinate. Since each function shares the same sign as its reciprocal function (sine and cosecant, cosine and secant, and tangent and cotangent), by remembering the signs for sine and cosine in the four quadrants and that tangent is the ratio of sine divided by cosine, you will automatically know the corresponding sign for secant, cosecant, and cotangent. Instead of using the Pythagorean identity sin 2 cos2 1, the problems can also be worked by drawing a picture of the situation. Then from the picture, it is a matter of using the trigonometric definitions and reading the values from the picture. Quadrant II (x, y) (-, +) Quadrant I (x, y) (+, +) Quadrant III (x, y) (-, -) Quadrant IV (x, y) (+, -) cos 0 sin 0 cos 0 sin 0 cos 0 sin 0 cos 0 sin 0 The Pythagorean Identities The equation of the circle pictured at the bottom of page 1 is x 2 y 2 r 2 . We know by the definition for sine and cosine that x y cos or x r cos and sin or y r sin . r r Substituting into the circle equation, we have 2 2 r cos r sin r 2 cos2 r 2 sin 2 r 2 cos2 sin 2 r 2 . ©2014;Dr. B. Shryock Page 3 of 7 Math130: Section 5.2 – Trigonometric Functions of Angles Since r 0, we can divide by r 2 , and the result is the Pythagorean identity sin 2 cos2 1. Notice that if we solve for either trigonometric term, this identity can be written as sin 1 cos2 or cos 1 sin 2 . Dividing sin 2 cos2 1 by sin 2 , the result is as follows. sin 2 cos 2 1 2 sin sin 2 1 cot 2 csc 2 Dividing sin 2 cos2 1 by cos 2 , the result is as follows. sin 2 cos 2 1 2 cos cos 2 . tan 1 sec 2 2 In summary, the Pythagorean Identities are given below. cos 2 sin 2 1 1 tan 2 sec 2 cot 2 1 csc2 We can work with factoring trigonometric expressions as in algebra. Example 2: Factor and simplify. sin 4 x cos 4 x Solution: Factoring sin 4 x cos 4 x is like factoring a 4 b4 in algebra with just a few slight differences. a 4 b4 (a 2 b 2 )(a 2 b2 ) (a 2 b 2 )(a b)(a b) sin 4 x cos 4 x (sin 2 x cos 2 x)(sin 2 x cos 2 x) (1)(sin 2 x cos 2 x) sin 2 x cos 2 x (sin x cos x)(sin x cos x) Trigonometric Identities Establishing an identity is basically proving that a trigonometric equation is true for all values of the argument. You may work on only one side of the identity to achieve the other given side. Even though you may not work on both sides, you may certainly look to see what you are trying to prove which may provide you with clues as to the next step in reaching your goal. ©2014;Dr. B. Shryock Page 4 of 7 Math130: Section 5.2 – Trigonometric Functions of Angles In general, begin by working on the more difficult side. Convert to the sine and cosine functions if you do not see another technique to use. Be familiar with the basic identities listed on page 338; these are the main tools for verifying other, more complex identities. You will want to memorize these since they will NOT be provided on the formula sheet for the test or the final exam. Be Patient and Persistent! If things are not working out, sometimes it is necessary to start over with a different plan, or begin again by working on the other side. Example 2 on page 336 shows the derivation for the 30, 45, and 60 angles. A chart summarizing the findings is at the bottom of the same page. Also refer to the Unit Circle you printed for Section 5.1. The sooner you familiarize yourself with these values, the easier you will find this course. Please watch/re-watch the Special Angles Video. Try to visualize this information instead of simply memorizing. Example 3: Evaluate cos 45 sin 30 tan 2 60 . Solution: cos 45 sin 30 tan 2 60 21 2 2 3 21 3 2 2 2 5 2 2 2 5 2 4 Pay close attention to your use of notation. tan tan 2 tan 2 2 Many applications of right triangles involve the angle of elevation and the angle of depression. The angle of elevation (see Figure 1 below) refers to the angle measured from the horizontal upward to the object in question. The angle of depression (see Figure 2 below) refers to the angle measured from the horizontal downward to the object. object observer horizontal angle of depresession angle of elevation observer Figure 1 horizontal Figure 2 object Example 4: A person is standing 60 m from the base of a tower. The angle of elevation to the top of the tower is 72. To the nearest tenth, how high is the tower? ©2014;Dr. B. Shryock Page 5 of 7 Math130: Section 5.2 – Trigonometric Functions of Angles Solution: The right triangle depicting the application is shown below. The height of the tower is labeled x, so the equation to use is as follows. x tan 72 60 T x 60 tan 72 184.7 O x W E R The tower is approximately 184.7 m high. 72° 60 m Example 5: A shoot from a 3rd floor apartment renovation has an angle of depression of 60. If the top of the shoot is 11 meters above level ground, how long is the shoot? Solution: The 60 angle of depression is the top angle in the depiction below. Recall from geometry that if two parallel lines are cut by a transversal, alternate interior angles are congruent. Since the angle of depression is 60, the base angle is also 60 because they are alternate interior angles. 11 L 11 L sin 60 11 22 L 3 3 2 sin 60 60° 11m L 60° The shoot is 22 m long. 3 Notice that this question did not say “round to the nearest …”, so we did not use our calculator and the answer is given in exact form. Also notice that it is fine to leave answers with a radical in the denominator provided the 6 2 6 2 6 fraction is simplified. For instance, is not simplified since 3 2. 2 2 2 2 Example 6: Find the exact value of sec if cot 3 and sin 0. Solution Begin by determining the quadrant of the terminal side of . If cot is positive, then tan is positive, and tan is positive in x 3 quadrants I and III. sin is negative in quadrants III and IV. Hence, the terminal side of is in quadrant III. y 1 1 1 y 10 . Since is in quadrant III and if cot 3 , then tan 3 3 x Sketch a right triangle in quadrant III as shown below labeling the sides of the triangle. Now find the length of the hypotenuse and insert that value into your sketch. ©2014;Dr. B. Shryock Page 6 of 7 Math130: Section 5.2 – Trigonometric Functions of Angles (3) 2 (1) 2 r 2 9 1 r2 r 2 10 r 10 Recall that sec is the reciprocal of cos . From the diagram, we see that cos sec x 3 , hence r 10 10 . 3 Practice Problems Be sure to work the Not on WA in the list below, but do not submit them for grading. They could appear on tests and the final exam. Answers to odd-numbered problems can be found at the end of your text. Section Problems on WA 5.2 6, 9, 12, 16, 20, 21, 24, 25, 26, 31, 37, 39, 43, 46, 75, 77, 86, 88, 90, 92, 94 ©2014;Dr. B. Shryock Not on WA 55, 57, 59, 63, 65, 67 Page 7 of 7