Problem 48 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \( \sin(A + B) = \sin A \cos B + \cos A \sin B \)

Step 2: Applying the Identity

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

Step 3: Solving the Expression

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