Problem 25 - Precalculus Essentials

Question

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\sin 25^{\circ} \cos 5^{\circ}+\cos 25^{\circ} \sin 5^{\circ}$$

Step-by-Step Solution

Answer

The given expression simplifies to \( \sin(30^{\circ}) = 0.5 \)

Step 1: Recognize the identity

Note that the given expression is of the form \(\sin A \cos B + \cos A \sin B\), which is the formula for \(\sin(A+B)\), where A = 25 degrees and B = 5 degrees.

Step 2: Apply the sine sum identity

Applying the sine sum identity to the expression, we get \(\sin(A+B) = \sin(25^{\circ}+5^{\circ}) = \sin(30^{\circ})\)

Step 3: Calculate the sine value

Next, find the exact value of the sine of 30 degrees, which according to the unit circle, is 0.5.