Pre-Calc Exam Notes 71

# Pre-Calc Exam Notes 71 - = A in Figure 3.2.1(a and ∠ QPR...

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Sum and Difference Formulas Section 3.2 71 3.2 Sum and Difference Formulas We will now derive identities for the trigonometric functions of the sum and difference of two angles. For the sum of any two angles A and B , we have the addition formulas : sin ( A + B ) = sin A cos B + cos A sin B (3.12) cos ( A + B ) = cos A cos B sin A sin B (3.13) To prove these, ±rst assume that A and B are acute angles. Then A + B is either acute or obtuse, as in Figure 3.2.1. Note in both cases that QPR = A , since QPR = QPO OPM = (90 B ) (90 ( A + B ))
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Unformatted text preview: = A in Figure 3.2.1(a), and ∠ QPR = ∠ QPO + ∠ OPM = (90 ◦ − B ) + (90 ◦ − (180 ◦ − ( A + B ))) = A in Figure 3.2.1(b). B O Q N P M R A A A + B (a) A + B acute O Q N P M R A A A + B (b) A + B obtuse Figure 3.2.1 sin ( A + B ) and cos ( A + B ) for acute A and B Thus, sin ( A + B ) = MP OP = MR + RP OP = NQ + RP OP = NQ OP + RP OP = NQ OQ · OQ OP + RP PQ · PQ OP = sin A cos B + cos A sin B , (3.14)...
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