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Sullivan Precalculus: Section 5.2 Trig Functions: Unit Circle Objectives of this Section • Find the Exact Value of the Trigonometric Functions Using the Unit Circle •Find the Exact Value of the Trigonometric Functions of 45 Degrees • Find the Exact Value of the Trigonometric Functions of 30 Degrees and 60 Degrees • Use a Calculator to Approximate the Value of the Trigonometric Functions of Acute Angles The unit circle is a circle whose radius is 1 and whose center is at the origin. Since r = 1: s r becomes s (0, 1) y s x (1, 0) (-1, 0) (0, -1) (0, 1) y t P = (a, b) (-1, 0) (1, 0) (0, -1) x Let t be a real number and let P = (a, b) be the point on the unit circle that corresponds to t. The sine function associates with t the y-coordinate of P and is denoted by sint b The cosine function associates with t the x-coordinate of P and is denoted by cost a If a 0 , the tangent function is defined as b tan t a If b 0 , the cosecant function is defined as 1 csc t b If a 0 , the secant function is defined as 1 sect a If b 0 , the cotangent function is defined as a cot t b 1 15 be Let t be a real number and let P , 4 4 the point on the unit circle that corresponds to t. Find the exact value of the six trigon ometric functions. a , b 1 4 , 15 sint b 4 15 4 1 cost a 4 a , b 1 , 4 15 b 4 tan t 1 a 4 1 1 csct b 15 4 1 1 sect 4 a 1 4 1 a 4 cot t b 15 4 15 4 15 4 4 15 15 15 1 15 15 15 (0, 1) y t P = (a, b) (-1, 0) (1, 0) (0, -1) x If t radians, the six trigonometric functions of the angle are defined as sin sin t cos cos t tan tan t csc csc t sec sec t cot cot t Find the exact value of the six trigonometric functions of 45 degrees. 45 c b=1 2 45 a=1 c a b 2 2 2 c 1 1 2 2 c 2 2 2 b 1 2 sin 45 sin 4 c 2 2 a 1 2 cos 45 cos 4 c 2 2 b 1 a 1 tan45 tan 1 cot 45 cot 1 4 a 1 4 b 1 c 2 csc 45 csc 2 4 b 1 c 2 sec 45 sec 2 4 a 1 Find the exact value of the six trigonometric functions of 30 and 60 degrees. 30 30 2 2 b 60 60 a a 2 1 b 2 2 2 b 2 1 4 1 3 b 3 2 2a = 2 so a = 1 c a b 2 2 2 2 2 a 1 b 3 sin 30 sin cos30 cos 6 c 2 6 c 2 sin 30 12 1 3 tan 30 tan 6 cos 30 32 3 3 1 1 cot 30 cot 3 6 tan 30 1 3 1 1 csc 30 csc 2 6 sin 30 1 2 1 1 2 2 3 sec 30 sec 6 cos 30 32 3 3 a 1 3 cos60 cos sin60 sin 3 c 2 3 c 2 sin 60 32 3 tan 60 tan 3 3 cos60 12 1 1 1 3 cot 60 cot 3 tan 60 3 3 1 1 2 2 3 csc 60 csc 3 sin 60 32 3 3 1 1 sec 60 sec 2 3 cos60 1 2 b Use a calculator to find the approximate value of (a) sin52 ( b) tan 5 (c) sec 5