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He who is devoid of the power to forgive is devoid of the power to love. Dr. Martin Luther King, Jr. (1929 – 1968) an American clergyman, activist, and prominent leader in the AfricanAmerican Civil Rights Movement Chapter 4 Trigonometry Day V Trigonometric Functions of Any Angle (4.4) You can use trigonometric functions to model and solve real life problems, like modeling the average daily temperature in a city. Goal 1 To evaluate trigonometric functions of any angle I. Introduction Life does not always give us unit circles, where r = 1. Why can’t r ≤ 0? Since r = any value, we have generalized trigonometry definitions Standard 2.1 Students know the definition of sine and cosine as y-and x-coordinates of points on the unit circle Standard 5.1 Students know the definitions of the tangent and cotangent functions. Standard 6.1 Students know the definitions of the secant and cosecant functions. (x, y) r = x² + y² y r sin = csc = , y 0 r y x r x0 cos = sec = , r x y x tan = , cot = , y 0 x y x0 Let’s play “Name that Quadrant” Name the Quadrants in which the sine function is positive. I and II sin cos tan sec csc cot I + __ __ __ __ __ __ II + __ __ __ __ __ __ III __ __ __ __ __ __ IV __ __ __ __ __ __ Name the Quadrants in which the sine function is negative. III and IV sin cos tan sec csc cot I + __ __ __ __ __ __ II + __ __ __ __ __ __ III – __ __ __ __ __ __ IV – __ __ __ __ __ __ Name the Quadrants in which the cosine function is positive. I and IV sin cos tan sec csc cot I + __ + __ __ __ __ __ II + __ __ __ __ __ __ III – __ __ __ __ __ __ IV – __ + __ __ __ __ __ Name the Quadrants in which the cosine function is negative. II and III sin cos tan sec csc cot I + __ + __ __ __ __ __ II + __ – __ __ __ __ __ III – __ – __ __ __ __ __ IV – __ + __ __ __ __ __ Name the Quadrants in which the tangent function is positive. I and III sin cos tan sec csc cot I + __ + __ + __ __ __ __ II + __ – __ __ __ __ __ III – __ – __ + __ __ __ __ IV – __ + __ __ __ __ __ Name the Quadrants in which the tangent function is negative. II and IV sin cos tan sec csc cot I + __ + __ + __ __ __ __ II + __ – __ – __ __ __ __ III – __ – __ + __ __ __ __ IV – __ + __ – __ __ __ __ Fill in the rest of the chart! sin cos tan sec csc cot I + __ + __ + __ + __ + __ + __ II + __ – __ – __ – __ + __ – __ III – __ – __ + __ – __ – __ + __ IV – __ + __ – __ + __ – __ – __ .8 over the 2. . . What’s your sine? Example 1 Evaluating Trigonometric Functions 1. Let (-12, -5) be a point on the terminal side of . Find the sine, cosine, and tangent of . -5 -12 y² r = x²+ ? = (-12)²+(-5)² = 169 = 13 1. Let (-12, -5) be a point on the terminal side of . Find the sine, cosine, and tangent of . -5 -12 13 sin = y/r = -5/13 cos = x/r =-12/13 tan = y/x = 5/12 2. Let (3, 1) be a point on the terminal side of . Find the sine, cosine, and tangent of . 3 1 y² r = x²+ ? = 3²+1² = 10 2. Let (3, 1) be a point on the terminal side of . Find the sine, cosine, and tangent of . 10 3 1 sin = y/r = 1/10 cos = x/r =3/10 tan = y/x = 1/3 Your Turn 1. Let (-5, 2) be a point on the terminal side of . Find the sine, cosine, and tangent of . 29 2 sin =y/r=2/29 -5 cos =x/r= -5/29 tan =y/x=2/-5 2. Let (3, -7) be a point on the terminal side of . Find the sine, cosine, and tangent of . 3 sin =y/r= -7/58 -7 cos =x/r= 3/58 58 tan =y/x=-7/3 sin 0 and cos 0 Where am I? sin 0 cos 0 III Your Turn sin 0 and tan 0 Where am I? sin > 0 tan 0 II sec 0 and cot 0 Where am I? sec > 0 cot 0 IV Example 2 Evaluating Trigonometric Functions If sin = ½ and tan 0, find the exact value of cos . Where is sin 0? Where is tan 0? is in Quadrant II. If sin = ½ and tan 0, find the exact value of cos . 1 2 -3 cos = x/r = -3/2 Your Turn 1. If cos = -4/5 and is in Quadrant II, find the exact value of sin . 5 3 -4 sin θ = 3/5 2. If csc = 4 and cot 0, find the exact value of cos . Where is csc 0? Where is cot 0? is in Quadrant II. 2. If csc = 4/1 and cot < 0, find the exact value of cos . 1 4 -15 cos θ = -15/4 What starts with T ends with T and is full of T? What starts with T ends with T and is full of T? Teapot A quadrantal angle is an angle that lies on the x- or y-axis. Example 3 Trigonometric Functions of Quadrantal Angles Find the cos of the four quadrantal angles. Unit circle cos = x/r (1 ? , ?0 ) cos 0 = 1/1 = 1 Find the cos of the four quadrantal angles. Unit circle (0 ? , 1? ) cos = x/r cos /2 = 0/1 = 0 Find the cos of the four quadrantal angles. Unit circle cos = x/r (-1 ?) ? ,0 cos = -1/1 = -1 Find the cos of the four quadrantal angles. cos = x/r Unit circle cos 3/2 = 0/-1 = 0 ( ?0 ,-1 ?) Your Turn If tan = undefined and ≤ θ ≤ 2, find the exact value of sin . tan θ = y/x? When is tan undefined? (0, -1) WhatSin θ isθ x==-1/1 0 between = -1 and 2? Goal 2 To use reference angles to evaluate trigonometric functions II. Reference Angles Standard 9.1 Students compute, by hand, the values of the trigonometric functions at various standard points. A reference angle is the acute angle formed by the terminal side of in standard position and the HORIZONTAL axis. Quadrant II = 180 - = - Quadrant III = - 180 = - Quadrant IV = 360 - = 2 - Example 4 Finding Reference Angles = 210 210 =210 - 180 = 30 = 4.1 3.14 4.1 = 4.1 - .96 4.71 Your Turn θ = 309 360 – 309 = 51 θ = -149 180 – 149 = 31 θ = 7/4 8 – 7 = 1 4 4 4 θ = 11/3 12 – 11 = 1 3 3 3 Example 7 Evaluate NO calculator - evaluate 2 sec θ = -2/1 -1 60 2 r = 2, x = -1 120 V 240 Where am I? 2/3 V 4/3 Your Turn NO calculator - evaluate sin θ = 2/2 22 1 1 sin θ= 1/2 45 y = 1, r = 2 45 V 135 Where am I? /4 V 3/4