Given:
A. [tex]$\sin (x)$[/tex]
B. [tex]$\cos (x)$[/tex]
C. [tex]$\cos (y)$[/tex]
D. [tex]$\sin (y)$[/tex]
Use the drop-down menus in the derivation of the cosine sum identity:
[tex]$\begin{array}{l}
sin (x+y) \
=\cos \left(\frac{\pi}{2}-(x+y)\right) \
=\cos \left(\left(\frac{\pi}{2}-x\right)-y\right) \
=\cos \left(\frac{\pi}{2}-x\right) \quad \hat{0}+\sin \left(\frac{\pi}{2}-x\right) \
=\sin (x) \cos (y)+\cos (x) \sin (y)
\end{array}$[/tex]
Answer (2)
Use the cosine subtraction formula: cos ( a − b ) = cos ( a ) cos ( b ) + sin ( a ) sin ( b ) .
Apply the formula to the expression cos (( 2 π − x ) − y ) with a = 2 π − x and b = y .
Use the identities cos ( 2 π − x ) = sin ( x ) and sin ( 2 π − x ) = cos ( x ) .
The missing terms are cos ( y ) and sin ( y ) .
Explanation
Problem Analysis We are given a partially completed derivation of the cosine sum identity and asked to fill in the missing terms. The derivation starts with the sine addition formula and uses the cofunction identity to rewrite it in terms of cosine. Then, the cosine subtraction formula is applied, and finally, the cofunction identities are used again to arrive at the final expression.
Filling in the missing terms The derivation is as follows:
sin ( x + y ) = cos ( 2 π − ( x + y )) = cos (( 2 π − x ) − y ) = cos ( 2 π − x ) cos ( y ) + sin ( 2 π − x ) sin ( y ) = sin ( x ) cos ( y ) + cos ( x ) sin ( y )
The missing terms are cos ( y ) and sin ( y ) .
Final Answer Therefore, the missing terms are C. cos ( y ) and D. sin ( y ) .
Examples
Understanding trigonometric identities like the cosine sum identity is crucial in fields like physics and engineering. For example, when analyzing wave interference, these identities help simplify complex expressions and predict the resulting wave patterns. In signal processing, they are used to decompose signals into their constituent frequencies, enabling efficient filtering and analysis.
The derivation of the sine addition formula leads to the final identity sin ( x + y ) = sin ( x ) cos ( y ) + cos ( x ) sin ( y ) . The missing terms in the given expression are cos ( y ) and sin ( y ) .
;
