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Section 10.4: Sum and Difference Formulas If I asked you to find an exact value of trigonometric function without a calculator, which angles could you use? The sum and difference formulas allow you to add or subtract any two of the special angles found on our unit circle WITHOUT a calculator. sin (u ± v) = sin u cos v ± cos u sin v (same sign) cos (u ± v) = cos u cos v ∓ sin u sin v (opposite sign) Ex. 1 Find the exact value of cos 75 (Ask yourself... how can I use the special angles from my unit circle to add or subtract and yield 75?) How about 45 + 30 ? cos 75 = cos 45 cos 30 - sin 45 sin 30 π. Ex. 2. Find the exact value of the sin12 (Ask yourself...how can I use the special angles from my unit circle to add or subtract and yield π/12? I know this involves fractions, but they are my friend.) Tip: This problem is in 12ths. So, look at the angles on your unit circle in terms of 12ths versus their reduced form. π π How about 3 - 4 ? π sin = 12 Ex. 3: Let’s Try A Tangent Problem! Find the exact value of tan 12 tan u tan v tan(u v) 1 tan u tan v tan tan 12 3 4 Ex. 4: Let’s go backward! Find the exact value of: sin 42 cos12 cos 42 sin12 10.4 Learning Opportunity Read Section 10.4 p. 653 #1-35 odd 10.5 Double and Half Angle Formulas In this section, we will continue to add to our identities. So far we have learned: Reciprocal Identities Quotient Identities Pythagorean Identities Cofunction Identities Even/Odd Identities Sum and Difference Identities Double Angle Formulas sin 2u 2sin u cos u cos 2u cos u sin u 2 2 2 cos 2 u 1 1 2sin 2 u 2 tan u tan 2u 1 tan 2 u Double Angle Formulas 1 cos sin 2 2 1 cos cos 2 2 The sign depends on the quadrant where θ/2 is located 1 cos sin tan 2 sin 1 cos Ex. 1 Use the following to find sin 2θ, cos 2θ, tan 2θ 5 3 cos , 2 13 2 Ex. 2 Use the following to find: sin , cos , tan 2 2 2 5 3 cos , 2 13 2 Ex. 3 Use the figure to find the exact value of the trigonometric function. a. tan θ b. sin 2θ c. sec 2θ 1 θ 4 d. cot 2θ Ex. 4 Use the figure to find the exact value of the trigonometric function. sin sec 2 2 8 θ 15 cot 2 10.5 Learning Opportunity Please Read Section 10.5 p. 660-661 #1-7 odd, 23-27 odd, 35-39 odd, 49-53 odd