sin u cos v
Fill in the blank to complete the trigonometric formula.
sin u cos v =  ÂÂ
The Correct Answer and Explanation is:
The correct trigonometric identity is:
$$
\sin u \cos v = \frac{1}{2}[\sin(u + v) + \sin(u – v)]
$$
Explanation (300+ words):
The expression $\sin u \cos v$ represents the product of the sine of one angle and the cosine of another. While this is a valid trigonometric expression by itself, it is often useful to rewrite such products as sums or differences of trigonometric functions—especially in calculus, signal processing, and solving trigonometric equations. This is where product-to-sum identities come into play.
The product-to-sum identities are a group of trigonometric formulas that convert the product of sine and cosine into the sum or difference of trigonometric functions. These identities are derived from the angle addition and subtraction formulas. For sine, the relevant identities are:
$$
\sin(u + v) = \sin u \cos v + \cos u \sin v
$$
$$
\sin(u – v) = \sin u \cos v – \cos u \sin v
$$
If we add these two identities:
$$
\sin(u + v) + \sin(u – v) = (\sin u \cos v + \cos u \sin v) + (\sin u \cos v – \cos u \sin v)
$$
The $\cos u \sin v$ terms cancel each other out:
$$
\sin(u + v) + \sin(u – v) = 2 \sin u \cos v
$$
Now divide both sides by 2:
$$
\sin u \cos v = \frac{1}{2}[\sin(u + v) + \sin(u – v)]
$$
This is the product-to-sum identity for $\sin u \cos v$.
Applications:
This identity is helpful when:
- Integrating products of sine and cosine.
- Solving trigonometric equations where products are harder to manipulate than sums.
- Simplifying wave functions in physics or engineering, especially in electrical engineering and acoustics.
For example, in signal processing, combining two signals often leads to expressions involving $\sin u \cos v$, and using this identity allows one to understand the frequencies present in the signal.
Thus, the identity:
$$
\sin u \cos v = \frac{1}{2}[\sin(u + v) + \sin(u – v)]
$$
is a powerful tool in both pure and applied mathematics
.