Pre-Calc Exam Notes 72

Pre-Calc Exam Notes 72 - ) cos B + cos ( A + 90 ) sin B ,...

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72 Chapter 3 Identities §3.2 and cos ( A + B ) = OM OP = ON MN OP = ON RQ OP = ON OP RQ OP = ON OQ · OQ OP + RQ PQ · PQ OP = cos A cos B sin A sin B . (3.15) So we have proved the identities for acute angles A and B . It is simple to verify that they hold in the special case of A = B = 0 . For general angles, we will need to use the relations we derived in Section 1.5 which involve adding or subtracting 90 : sin ( θ + 90 ) = cos θ sin ( θ 90 ) = − cos θ cos ( θ + 90 ) = − sin θ cos ( θ 90 ) = sin θ These will be useful because any angle can be written as the sum of an acute angle (or 0 ) and integer multiples of ± 90 . For example, 155 = 65 + 90 , 222 = 42 + 2(90 ), 77 = 13 90 , etc. So if we can prove that the identities hold when adding or subtracting 90 to or from either A or B , respectively, where A and B are acute or 0 , then the identities will also hold when repeatedly adding or subtracting 90 , and hence will hold for all angles. Replacing A by A + 90 and using the relations for adding 90 gives sin (( A + 90 ) + B ) = sin (( A + B ) + 90 ) = cos ( A + B ) , = cos A cos B sin A sin B (by equation (3.15)) = sin ( A + 90
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Unformatted text preview: ) cos B + cos ( A + 90 ) sin B , so the identity holds for A + 90 and B (and, similarly, for A and B + 90 ). Likewise, sin (( A 90 ) + B ) = sin (( A + B ) 90 ) = cos ( A + B ) , = (cos A cos B sin A sin B ) = ( cos A ) cos B + sin A sin B = sin ( A 90 ) cos B + cos ( A 90 ) sin B , so the identity holds for A 90 and B (and, similarly, for A and B + 90 ). Thus, the addition formula (3.12) for sine holds for all A and B . A similar argument shows that the addition formula (3.13) for cosine is true for all A and B . QED Replacing B by B in the addition formulas and using the relations sin ( ) = sin and cos ( ) = cos from Section 1.5 gives us the subtraction formulas : sin ( A B ) = sin A cos B cos A sin B (3.16) cos ( A B ) = cos A cos B + sin A sin B (3.17)...
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