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Cosines

# Cosines - Rewriting B cos(t t0 C sin(t t0 as A cos(t t1 In...

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Rewriting B cos( ω ( t - t 0 )) + C sin( ω ( t - t 0 )) as A cos( ω ( t - t 1 )) In several important applications, we will ﬁnd that we get solutions to dif- ferential equations in the form B cos( ω ( t - t 0 )) + C sin( ω ( t - t 0 )) where B , C and T 0 are constants. In order to be able to clearly visualize and easily understand the behavior of such functions, it is very useful to be able to write them in the form A cos( ω ( t - t 1 )) In that form it is easy to see what the maximum and minimum values are, and also for which values of t the maximum and the minimum are reached. This rewriting is based on a trigonometric identity: cos( θ - φ ) = cos( φ ) cos( θ ) + sin( φ )sin( θ ) We can use that identity to try to match Acos ( θ - φ ) with B cos( θ )+ C sin( θ ). We see that Acos ( θ - φ ) = A cos( φ )cos( θ ) + A sin( φ )sin( θ ) , so if we can solve A cos( φ ) = B and A sin( φ ) = C We will have Acos ( θ - φ ) = B cos( θ ) + C sin( θ ) . We can solve for A and φ by looking at a right triangle with base B , and

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Cosines - Rewriting B cos(t t0 C sin(t t0 as A cos(t t1 In...

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