Use the product-to-sum formula to find the exact value:

If $\cos 37.5^\circ \sin 7.5^\circ = \frac{\sqrt{A} - B}{4}$, then:

$A = \_\_$ ,

$B = \_\_$ .

Neetesh Kumar | October 16, 2024

Pre-Calculus Homework Help

Sum-to-Product and Product-to-Sum Formulas Assignment 9.4 Question 03
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Step-by-step solution:

We will use the product-to-sum formula for $\cos(A)\sin(B)$:

Trignometric-ratio Formula Sheet

$\cos(A)\sin(B) = \frac{1}{2}[\sin(A + B) - \sin(A - B)]$

Given expression: $\cos 37.5^\circ \sin 7.5^\circ$

Using the product-to-sum formula:

$\cos 37.5^\circ \sin 7.5^\circ = \frac{1}{2} [\sin(37.5^\circ + 7.5^\circ) - \sin(37.5^\circ - 7.5^\circ)]$

Simplify:

$= \frac{1}{2} [\sin(45^\circ) - \sin(30^\circ)]$

We know that:

$\sin(45^\circ) = \frac{\sqrt{2}}{2}$

$\sin(30^\circ) = \frac{1}{2}$

Thus:

$= \frac{1}{2} \left(\frac{\sqrt{2}}{2} - \frac{1}{2}\right)$

$= \frac{1}{2} \cdot \frac{\sqrt{2} - 1}{2}$

$= \frac{\sqrt{2} - 1}{4}$

This matches the given equation:

$\frac{\sqrt{A} - B}{4} = \frac{\sqrt{2} - 1}{4}$

From this, we can see that:

$A = 2$

$B = 1$

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Final answers:

$A = \boxed{2}, B = \boxed{1}$

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