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Ch. 5 – Analytic Trigonometry 5.4 – Sum and Difference Formulas Angle sum and difference formulas can be used to find exact values of trig functions not on the unit circle. Ex: Find the exact value of cos75°. ◦ ◦ ◦ ◦ ◦ 75° isn’t on the unit circle, but 30° and 45° are, and 75 = 30 + 45! Instead of finding cos75°, find cos(30° + 45°)! Look up your angle sum formula… cos(a + b) = cosa cosb – sina sinb cos(30 + 45) = cos30 cos45 – sin30 sin45 3 2 1 2 2 2 2 2 6 2 4 4 6 2 4 Ex: Find the exact value of sin(π/12). ◦ Convert to degrees if you like! ◦ Find angles that either add or subtract to π/12… 3 4 ◦ Look up your angle sum formula… ◦ sin(a – b) = sina cosb – sinb cosa sin sin cos sin cos 3 4 4 3 3 4 3 2 21 2 2 2 2 6 2 4 4 6 2 4 Ex: Write cos(arctan 1 + arccos x) as an algebraic expression. ◦ This is like an angle sum problem! ◦ Draw a right triangle for each inverse trig function. Label the angles differently. ◦ Use Pythagorean Thm. to complete the triangles… 2 1 1 x2 1 u v x 1 ◦ We’re just finding cos(u + v) = cosu cosv – sinu sinv 1 x 1 1 x2 1 21 2 x 1 x2 2 x 1 x2 2 2 I’ll let you leave the radical in the denominator only for these algebraic answers. Find the exact value of cos(-15°). 1. 6 2 4 2. 3. 6 2 2 4. 5. 6 2 2 6 2 4 6 2 4 0% 0% 0% 0% 0% Simplify tan(θ – 3π). 1. tan 2. 3. tan 1 tan 4. 5. undefined tan 1 tan tan 1 1 tan 0% 0% 0% 0% 0% Simplify sin(x + x). 1. 1 2 2 2. sin x cos x 3. sin x cos x 4. 0 5. 2 2 2sin x cos x 0% 0% 0% 0% 0% Ex: Angles u and v both lie in quadrant I. If sinu = 12/13 and cosv = 3/5, find sin(u+v). ◦ No angles are given this time, just trig ratios…and not even trig ratios from the unit circle! ◦ Draw some triangles: 5 13 12 u 4 v 5 3 ◦ Use Pythagorean Theorem to get the other side lengths ◦ To get sin(u+v), use an angle sum formula: sin(u v) sin u cos v sin v cos u 36 20 12 3 4 5 56 65 65 13 5 5 13 65 Ex: Angles u and v both lie in quadrant II. If tanu = -7/24 and sinv = 8/17, find cos(u-v). ◦ Draw some triangles, but think…the adjacent side should be negative in both triangles because both angles lie in the 2nd quadrant: 25 u 17 7 v 24 15 ◦ Use Pythagorean Theorem to get the other side lengths ◦ To get cos(u-v), use an angle sum formula: cos(u v) cos u cos v sin u sin v 24 15 7 8 25 17 25 17 360 56 416 425 425 425 8