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4.4 Trigonometric Functions of Any Angle Objectives Evaluate trigonometric functions of any angle. Find reference angles. Evaluate trigonometric functions of real numbers. 2 Introduction 3 Introduction Recall that when using the unit circle to evaluate the value of a trig function, cos θ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig values are really cos θ = x/1 and sin θ = y/1. So what if the radius (hypotenuse) is not 1? 4 Introduction The definitions of trigonometric functions were restricted to acute angles. In this section, the definitions are extended to cover any angle. When is an acute angle, the definitions here coincide with those given in the preceding section. 5 Introduction Because r = cannot be zero, it follows that the sine and cosine functions are defined for any real value of . However, when x = 0, the tangent and secant of are undefined. For example, the tangent of 90 is undefined. Similarly, when y = 0, the cotangent and cosecant of are undefined. 6 Introduction The previous definitions imply that tan θ and sec θ are not defined when x = 0. So what values of θ are we talking about? 3 , or 90 ̊, 270 ̊ 2 2 They also imply that cot θ and csc θ are not defined when y = 0. So what values of θ are we talking about? 0, or 0 ̊, 180 ̊ 7 Example – Evaluating Trigonometric Functions Let (–3, 4) be a point on the terminal side of . Find the sine, cosine, and tangent of . Solution: You can see that x = –3, y = 4, and 4 -3 8 Example – Solution cont’d So, you have the following. 4 5 -3 9 Trig of Any Angle Let (x, y) be a point on the terminal side of an angle θ in standard position with 2 2 r x y 0 y sin y r csc cos x r r , x0 x x cot , y 0 y y tan , x 0 x r , y0 y sec x 10 Your Turn: Let θ be an angle whose terminal side contains the point (−2, 5). Find the six trig functions for θ. sin 5 29 cos tan 2 29 5 2 csc sec cot 29 5 29 2 2 5 11 Trig of Any Angle The signs of the trigonometric functions in the four quadrants can be determined from the definitions of the functions. For instance, because cos = x/r, it follows that cos is positive wherever x 0, which is in Quadrants I and IV. (Remember, r is always positive.) 12 Trig of Any Angle In a similar manner, you can verify the results shown. 13 Example: Given sin θ = 4/5 and tan θ < 0, find cos θ and csc θ. x 52 42 3 cos csc 3 5 5 4 5 4 -3 14 Your Turn: Given tan θ = -3/2 and sin θ < 0, find cos θ and csc θ. Solution: Quadrant IV – cos positive & csc negative r 13 3 = 32 + 22 = 13 2 2 cos = = 13 2 13 cos = 13 13 csc = − = − 3 13 csc = − 3 15 Reference Angles 16 Reference Angles The values of the trigonometric functions of angles greater than 90 (or less than 0) can be determined from their values at corresponding acute angles called reference angles. ’ 17 Reference Angles The reference angles for in Quadrants II, III, and IV are shown below. ′ = – (radians) ′ = 180 – (degrees) ′ = – (radians) ′ = – 180 (degrees) ′ = 2 – (radians) ′ = 360 – (degrees) 18 Example – Finding Reference Angles Find the reference angle ′. a. = 300 b. = 2.3 c. = –135 19 Example (a) – Solution Because 300 lies in Quadrant IV, the angle it makes with the x-axis is ′ = 360 – 300 = 60. Degrees The figure shows the angle = 300 and its reference angle ′ = 60. 20 Example (b) – Solution cont’d Because 2.3 lies between /2 1.5708 and 3.1416, it follows that it is in Quadrant II and its reference angle is ′ = – 2.3 0.8416. Radians The figure shows the angle = 2.3 and its reference angle ′ = – 2.3. 21 Example (c) – Solution cont’d First, determine that –135 is coterminal with 225, which lies in Quadrant III. So, the reference angle is ′ = 225 – 180 = 45. Degrees The figure shows the angle = –135 and its reference angle ′ = 45. 22 Reference Angles When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2. 23 Your Turn: Find the reference angle for each of the following. 1. 213° 2. 1.7 rad 3. −144° 213 180 33 1.7 1.44 -144 ̊ is coterminal to 216 ̊ 216 ̊ - 180 ̊ = 36 ̊ 24 Trigonometric Functions of Real Numbers 25 Trigonometric Functions of Real Numbers To see how a reference angle is used to evaluate a trigonometric function, consider the point (x, y) on the terminal side of , as shown in figure below. opp = | y |, adj = | x | 26 Trigonometric Functions of Real Numbers How Reference Angles Work: sin y r sin ' y r Same except maybe a difference of sign, depending on the quadrant the terminal side of is in. 27 Trigonometric Functions of Real Numbers To find the value of a trig function of any angle: 1. Find the trig value for the associated reference angle. 2. Pick the correct sign depending on where the terminal side lies. 28 Trigonometric Functions of Real Numbers So, it follows that sin and sin ′ are equal, except possibly in sign. The same is true for tan and tan ′ and for the other four trigonometric functions. In all cases, the quadrant in which lies determines the sign of the function value. 29 Trigonometric Functions of Real Numbers You can greatly extend the scope of exact trigonometric values. For instance, knowing the function values of 30 means that you know the function values of all angles for which 30 is a reference angle. 30 Trigonometric Functions of Real Numbers For convenience, the table below shows the exact values of the sine, cosine, and tangent functions of special angles and quadrant angles. Trigonometric Values of Common Angles 31 Example – Using Reference Angles Evaluate each trigonometric function. a. cos b. tan(–210) c. csc 32 Example (a) – Solution Because = 4 /3 lies in Quadrant III, the reference angle is as shown in the figure. Moreover, the cosine is negative in Quadrant III, so 33 Example (b) – Solution cont’d Because –210 + 360 = 150, it follows that –210 is coterminal with the second-quadrant angle 150. So, the reference angle is ′ = 180 – 150 = 30, as shown in the figure. 34 Example (b) – Solution cont’d Finally, because the tangent is negative in Quadrant II, you have tan(–210) = (–) tan 30 = . 35 Example (c) – Solution cont’d Because (11 /4) – 2 = 3 /4, it follows that 11 /4 is coterminal with the second-quadrant angle 3 /4. So, the reference angle is ′ = – (3 /4) = /4, as shown in the figure. 36 Example (c) – Solution cont’d Because the cosecant is positive in Quadrant II, you have 37 Your Turn: Evaluate: 1. sin 5/3 3 2 2. cos (−60°) 1 2 3. tan 11/6 3 3 38 Your Turn: Let θ be an angle in Quadrant III such that sin θ = −5/13. Find a) sec θ and b) tan θ using trig identities. sin cos 1 2 2 13 sec = − 12 2 5 2 cos 1 13 25 169 144 cos 2 169 12 12 cos , Q III cos 13 13 cos2 1 5 5 tan 13 12 12 13 39 Assignment: Pg. 294-296: #1 – 107 odd, 111. 40