find the exact value using a sum or difference identity. sin 75° √2 + √? / hint: sin(a ± b) = sin a cos b ± cos a sin b cos(a ± b) = cos a cos b ∓ sin a sin b

Question

find the exact value using a sum or difference identity. sin 75° √2 + √? /   hint: sin(a ± b) = sin a cos b ± cos a sin b cos(a ± b) = cos a cos b ∓ sin a sin b

Accepted Answer

# Explanation:
## Step1: Express $75^{\circ}$ as a sum of two angles
We know that $75^{\circ}=45^{\circ}+30^{\circ}$

## Step2: Apply the sine sum identity $\sin(A + B)=\sin A\cos B+\cos A\sin B$
Here $A = 45^{\circ}$ and $B=30^{\circ}$
$$
\begin{align*}
\sin(45^{\circ}+ 30^{\circ})&=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}\
&=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}\
&=\frac{\sqrt{6}+\sqrt{2}}{4}\
&=\frac{\sqrt{2}+\sqrt{6}}{4}
\end{align*}
$$

# Answer:
The green box (numerator - under the second square - root) is $6$ and the gray box (denominator) is $4$

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QUESTION IMAGE

Question

Explanation:

Step1: Express \(75^{\circ}\) as a sum of two angles

Step2: Apply the sine sum identity \(\sin(A + B)=\sin A\cos B+\cos A\sin B\)

Answer: