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NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 12-3 Study Guide and Intervention Trigonometric Functions of General Angles Trigonometric Functions for General Angles Trigonometric Functions, θ in Standard Position Let θ be an angle in standard position and let P(x, y) be a point on the terminal side of θ. By the Pythagorean Theorem, the distance r from the origin is given by r = √𝑥 2 + 𝑦 2 . The trigonometric functions of an angle in standard position may be defined as follows. sin θ = csc θ = 𝑦 cos θ = 𝑟 𝑟 𝑦 𝑥 𝑟 ,y≠0 𝑦 tan θ = 𝑟 𝑥 ,x≠0 𝑥 sec θ = , x ≠ 0 cot θ = , y ≠ 0 𝑥 𝑦 Example: Find the exact values of the six trigonometric functions of θ if the terminal side of θ in standard position contains the point ( –5, 5√𝟐). You know that x = –5 and y = 5. You need to find r. r = √𝑥 2 + 𝑦 2 Pythagorean Theorem = √(−5)2 + (5√2)2 Replace x with –5 and y with 5√2 . = √75 or 5√3 Now use x = –5, y = 5√2 , and r = 5√3 to write the six trigonometric ratios. 𝑦 5√2 5√3 = √6 3 cos θ = 𝑟 = 5 𝑟 5√3 5√2 = √6 2 sec θ = 𝑥 = sin θ = 𝑟 = csc θ = 𝑦 = 𝑥 −5 √3 𝑟 5√3 −5 = − √3 3 = −√3 𝑦 5√2 −5 = −√2 𝑥 −5 √2 = − tan θ = 𝑥 = cot θ = 𝑦 = 5 √2 2 Exercises The terminal side of θ in standard position contains each point. Find the exact values of the six trigonometric functions of θ. 1. (8, 4) 2. (4, 4) 3. (0, 4) 4. (6, 2) Chapter 12 17 Glencoe Algebra 2 NAME _____________________________________________ DATE ____________________________ PERIOD _____________ 12-3 Study Guide and Intervention (continued) Trigonometric Functions of General Angles Trigonometric Functions with Reference Angles If θ is a nonquadrantal angle in standard position, its reference angle θ ' is defined as the acute angle formed by the terminal side of θ and the x-axis. Reference Angle Rule Example 1: Sketch an angle of measure 205°. Example 2: Use a reference angle to find the exact Then find its reference angle. value of cos Because the terminal side of 205° lies in Quadrant III, the reference angle θ ' is 205° – 180° or 25°. Because the terminal side of 𝟑𝝅 . 𝟒 3𝜋 lies in 4 3𝜋 𝜋 or . 4 4 Quadrant II, the reference angle θ ' is π – The cosine function is negative in Quadrant II. cos 3𝜋 4 𝜋 = – cos 4 = – √2 2 Exercises Sketch each angle. Then find its reference angle. 1. 155° 2. 230° 3. 4𝜋 3 4. – 𝜋 6 Find the exact value of each trigonometric function. 5. tan 330° Chapter 12 6. cos 11𝜋 4 7. cot 30° 18 𝜋 8. csc 4 Glencoe Geometry