Review the incomplete derivation of the cosine sum identity.
| Step 1 | [tex]\cos (x+y)[/tex] |
| :----- | :---------------------------------- |
| Step 2 | [tex]$\sin \left(\frac{\pi}{2}-(x+y)\right)$[/tex] |
| Step 3 | |
| Step 4 | [tex]$\sin \left(\frac{\pi}{2}-x\right) \cos (y)-\cos \left(\frac{\pi}{2}-x\right) \sin (y)$[/tex] |
| Step 5 | |
Which expressions for Step 3 and Step 5 complete the derivation?
A. Step 3: [tex]$\sin \left(\left(\frac{\pi}{2}-x\right)+y\right)$[/tex]
Step 5: [tex]$\cos (x) \cos (y)+\sin (x) \sin (y)$[/tex]
B. Step 3: [tex]$\sin \left(\left(\frac{\pi}{2}-x\right)+y\right)$[/tex]
Step 5: [tex]$\cos (x) \cos (y)-\sin (x) \sin (y)$[/tex]
C. Step 3: [tex]$\sin \left(\left(\frac{\pi}{2}-x\right)-y\right)$[/tex]
Step 5: [tex]$\cos (x) \cos (y)+\sin (x) \sin (y)$[/tex]
D. Step 3: [tex]$\sin \left(\left(\frac{\pi}{2}-x\right)-y\right)$[/tex]
Step 5: [tex]$\cos (x) \cos (y)-\sin (x) \sin (y)$[/tex]
Answer (2)
Step 2 uses the identity cos ( θ ) = sin ( 2 π − θ ) .
Step 3 rewrites the argument as ( 2 π − x ) − y .
Step 4 applies the sine difference identity.
Step 5 simplifies the expression using cofunction identities, resulting in cos ( x ) cos ( y ) − sin ( x ) sin ( y ) .
Explanation
Understanding the Problem We are given an incomplete derivation of the cosine sum identity and need to find the missing expressions for Step 3 and Step 5. The derivation starts with cos ( x + y ) and aims to transform it into an expression involving cos ( x ) , cos ( y ) , sin ( x ) , and sin ( y ) .
Applying the Cofunction Identity Step 1: cos ( x + y )
Step 2: sin ( 2 π − ( x + y ))
This step uses the identity cos ( θ ) = sin ( 2 π − θ ) , where θ = x + y .
Rewriting the Argument Step 3: We need to rewrite sin ( 2 π − ( x + y )) in a form that allows us to use the sine difference identity. We can rewrite the argument as ( 2 π − x ) − y . Therefore, Step 3 is sin (( 2 π − x ) − y ) .
Applying the Sine Difference Identity Step 4: sin ( 2 π − x ) cos ( y ) − cos ( 2 π − x ) sin ( y )
This step applies the sine difference identity: sin ( a − b ) = sin ( a ) cos ( b ) − cos ( a ) sin ( b ) , where a = 2 π − x and b = y .
Simplifying the Expression Step 5: We need to simplify the expression from Step 4 using the cofunction identities: sin ( 2 π − x ) = cos ( x ) and cos ( 2 π − x ) = sin ( x ) . Substituting these identities into the expression from Step 4, we get: cos ( x ) cos ( y ) − sin ( x ) sin ( y ) . Therefore, Step 5 is cos ( x ) cos ( y ) − sin ( x ) sin ( y ) .
Examples
The cosine sum identity is used in physics to analyze wave interference and in signal processing to decompose complex signals into simpler components. For example, when two sound waves with frequencies f 1 and f 2 interfere, the resulting wave's amplitude depends on cos ( 2 π ( f 1 + f 2 ) t ) , which can be expanded using the cosine sum identity to analyze the individual frequency components. This helps in understanding how different frequencies combine to create complex sounds or signals.
The derivation for the cosine sum identity can be completed by expressing Step 3 as sin ( ( 2 π − x ) − y ) and Step 5 as cos ( x ) cos ( y ) − sin ( x ) sin ( y ) . Hence, the correct choice is D.
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