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3.4 Circular Functions The key trigonometric functions, sine and cosine, may be defined using the unit circle, which is symmetric with respect to the x-axis, the y-axis, and the origin. The equation of the unit circle is x2 + y2 = 1. y x2 + y2 = 1 x (1,0) The terminal side of an angle in standard position intersects the unit circle at a unique point P(x, y). The x-value of this point is defined to be cosine The y-value of this point is defined to be sine Because cos and sin are defined using the unit circle, they are called circular functions. -1 < cos < 1 and -1 < sin < 1 Example 1: Use the definitions of the sine and cosine functions to find each value. y x 𝜋 𝜋 (a) cos 2 (b) sin 2 𝜋 (c) cos (− 2 ) 𝜋 (d) sin (− 2 ) Advanced Mathematics/Trigonometry: 3.4 Circular Functions Page 1 For any angle , the values of cos and sin can be found if a point Q(x, y) on the terminal side is known, even if Q is not on the unit circle. This can be done using a reference triangle, which is found by drawing a perpendicular from Q to the x-axis. y Q(x, y) r y x x Let be an angle in standard position, and let (x, y) be a point distinct from the origin on the terminal side of , then: cos 𝜃 = 𝑥 𝑟 and sin 𝜃 = 𝑦 𝑟 where 𝑟 = √𝑥 2 + 𝑦 2 Example 2: Draw the reference triangle and find the exact values of cos and sin of the terminal side of an angle in standard position passes through each point. (a) P(2, 5) (b) Q(3, 7) 1 (c) R(2, 2) Advanced Mathematics/Trigonometry: 3.4 Circular Functions Page 2 Example 3: Find the exact values of cos and sin for angle in standard position with the given point on its terminal side. Draw the reference triangle. (a) P(-3, 4) 1 (b) Q(-2, -2) (c) R(-6, 9) 2 (d) S(-1, -5) Advanced Mathematics/Trigonometry: 3.4 Circular Functions Page 3 Since r, which is sometimes called the radius vector is always positive, the signs of cos and sin are determined by the signs of x and y. Therefore these signs depend on the quadrant in which the terminal side lies. y II I (+,+) (-,+) x (-,-) (+,-) III I Quadrant IV II III IV cos + - - + sin + + - - Example 4: State whether each value is positive, negative, or zero. (a) cos 45 (b) sin 3 (c) sin 5𝜋 4 (d) cos 75 (e) sin (-75) (f) cos 5 (g) sin (-3) Advanced Mathematics/Trigonometry: 3.4 Circular Functions Page 4 Example 5: (a) Angle is in standard position with its terminal side in the third quadrant. Find the exact value of cos 1 if sin is -2. (b) Angle is in standard position with its terminal side in the second quadrant. Find the exact value of 8 cos if sin is 10. (c) Angle is in standard position with its terminal side in the fourth quadrant. Find the exact value of 4 sin if cos is7. Homework: Day 1: p. 144 => Class Exercises 1 – 9; Day 2: p. 145 => Practice Exercises 1 – 22; 25 - 32 Advanced Mathematics/Trigonometry: 3.4 Circular Functions Page 5