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Chapter 4: Problem 48

Find the exact value of each expression. Do not use a calculator. $$\cos 12^{\circ} \sin 78^{\circ}+\cos 78^{\circ} \sin 12^{\circ}$$

Short Answer

Expert verified
The exact value of the expression is 1 because \( \sin 90^{\circ} = 1 \).

Step by step solution

01

Recognizing Trigonometric Identity

Notice that the two angles, 12 degrees and 78 degrees, add up to 90 degrees. Therefore, we can use the sine of sum of two angles identity: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)
02

Applying the Identity

We can rewrite the provided equation \( \cos 12^{\circ} \sin 78^{\circ} + \cos 78^{\circ} \sin 12^{\circ} \) as \( \sin(12^{\circ} + 78^{\circ}) \)
03

Solving the Expression

Now, solve \( \sin(12^{\circ} + 78^{\circ}) \) which equals \( \sin 90^{\circ} \).

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