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Math Analysis Day V. Trigonometric Functions of Any Angle (4.4) 10/11/11 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 African-American Civil Rights Movement 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. Why can’t r ≤ 0? 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. Since r = any value, we have generalized trigonometry definitions sin = csc = cos = sec = tan = cot = Let’s play “Name that Quadrant” I sin = cos = tan = sec = csc = cot = II III IV Example 1 - Evaluating Trigonometric Functions 1. Let (-12, -5) be a point on the terminal side of . Find the sine, cosine, and tangent of . 2. Let (3, 1) be a point on the terminal side of . Find the sine, cosine, and tangent of . Your Turn 1. Let (-5, 2) be a point on the terminal side of . Find the sine, cosine, and tangent of . 2. Let (3, -7) be a point on the terminal side of . Find the sine, cosine, and tangent of . Where am I? sin 0 and cos 0 Your Turn sin 0 and tan 0 sec 0 and cot 0 Example 2 - Evaluating Trigonometric Functions If sin = ½ and tan 0, find the exact value of cos . Your Turn 1. If cos = -4/5 and is in Quadrant II, find the exact value of sin . 2. If csc = 4/1 and cot < 0, find the exact value of cos . ______________________________________________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. Goal 2 - To use reference angles to evaluate trigonometric functions Standard 9.1 - Students compute, by hand, the values of the trigonometric functions at various standard points. II. Reference Angles ______________________________________is the acute angle formed by the ___________________ of in standard position and the ___________________________________________ axis. Quadrant II Quadrant III Quadrant IV Example 4 - Finding Reference Angles = 210 = 4.1 Your Turn 1. θ = 309 2. θ = -149 3. θ = 7/4 4. θ = 11/3