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4.3 Trigonometry Extended: The Circular Functions Objective: Find the trig function value of any angle. Explore trig functions as periodic functions. Explore and use the unit circle. WARMUP (1) The value of csc 5 . Find the values of the other five trig functions. 2 (2) A 12-foot ladder leans against the side of a house at an angle of 72 with the ground. How high up the house is the ladder? Drawing an Angle Coterminal Angles Coterminal angles are angles that _____________________________. There are an infinite number of angles that fit this description. For instance, 70 is coterminal with 430 AND 290 . Ex 1: Find one positive and one negative coterminal angle for each of the following angle measures: 305 Key-Points 132 SOH CAH TOA - Redefined The Big 3 Trig Ftns Old School New School Some Space for the Menorah sin cos tan cot sec csc Reciprocal Ftns Old School New School Evaluating Trig Functions Based on Location Ex 2: Find the values of all 6 trig functions if an angle, , is drawn through the point given. Key-Points ( 8, 6) ( 2 3, 2) Ex 3: Find the exact value of sec if the terminal side of an angle passes through the point (24, 7) . Find the exact value of sin if the terminal side of an angle passes through the point ( 3, 4) . Key-Points Understanding Signs The signs of an ordered pair, coupled with our trig function definitions hold valuable clues that can aid us in evaluating trig functions. Ex 4: Where is ? sin 0 tan 0 Key-Points cot 0 sin 0 sec 0 csc 0 THE UNIT CIRCLE Exploring the Unit Circle – With your group…. (1) For any value of t, the value of cos(t ) lies between _____ and _____ inclusive. (2) For any value of t, the value of sin(t ) lies between _____ and _____ inclusive. (3) The values of cos(t ) and cos( t ) are ___________________ each other, making this function even. (4) The values of sin(t ) and sin( t ) are ___________________ each other, making this function odd. (5) The values of sin(t ) and sin(t 2 ) are ___________________ each other. In fact, this is true of all of the six trig functions on their domains. (6) The values of sin(t ) and sin(t ) are ___________________ each other. The same is true for cos(t ) and cos(t ) . (7) The values of tan(t ) and tan(t ) are ___________________ each other, unless they are both undefined. (8) The sum sin 2 (t ) cos 2 (t ) is always equal to _____. Ex 5: Use your knowledge of the unit circle to evaluate the following: cos 225 sec 315 cot 5 tan 690 csc(420 ) Ex 6: Find the values of all 6 trig functions if an angle, , is drawn through the point on the unit circle given by: 2 2 1 , 3 3 Key-Points Using One Trig Ratio to Find the Others 5 Ex 7: Find the exact value of tan if csc and . 2 2 1 Find the exact value of cot if cos and csc 0 . 3 Find the exact value of cot if sec 5 and sin 0 . K-Points