Question Express $16 \sin (50 r) \sin (23 r)$ as a sum or difference of trigonometric functions.

Studdy Solution

STEP 1

Assumptions
1. We are given the expression $16 \sin (50r) \sin (23r)$ .
2. We need to write the product of the sine functions as a sum or a difference.
3. We will use the sine product-to-sum identities to convert the product into a sum or difference.

STEP 2

Recall the product-to-sum identities for sine functions:
$\sin A \sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$

STEP 3

Identify $A$ and $B$ from the given expression:
$A = 50r, \quad B = 23r$

STEP 4

Apply the product-to-sum identity to the given expression using the identified values of $A$ and $B$ :
$16 \sin (50r) \sin (23r) = 16 \cdot \frac{1}{2}[\cos(50r - 23r) - \cos(50r + 23r)]$

STEP 5

Simplify the expression inside the brackets by performing the subtraction and addition of $A$ and $B$ :
$\cos(50r - 23r) = \cos(27r)$ $\cos(50r + 23r) = \cos(73r)$

STEP 6

Substitute the simplified terms back into the expression:
$16 \sin (50r) \sin (23r) = 16 \cdot \frac{1}{2}[\cos(27r) - \cos(73r)]$

STEP 7

Multiply the constant factor $\frac{1}{2}$ by 16:
$16 \cdot \frac{1}{2} = 8$

STEP 8

Write the final expression as a sum or a difference:
$16 \sin (50r) \sin (23r) = 8[\cos(27r) - \cos(73r)]$
The product $16 \sin (50r) \sin (23r)$ is written as a difference $8[\cos(27r) - \cos(73r)]$ .

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