Problem 26 - Precalculus Essentials

Question

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\sin 40^{\circ} \cos 20^{\circ}+\cos 40^{\circ} \sin 20^{\circ}$$

Step-by-Step Solution

Answer

The expression \(\sin 40^{\circ} \cos 20^{\circ}+\cos 40^{\circ} \sin 20^{\circ}\) simplifies to \(\sin60^{\circ}\), which is equal to \(\frac{\sqrt{3}}{2}\).

Step 1: Identify the appropriate identity

Recognize the given expression \(\sin 40^{\circ} \cos 20^{\circ}+\cos 40^{\circ} \sin 20^{\circ}\) as a form of the trigonometric identity \(\sin(a) \cos(b) + \cos(a) \sin(b) = \sin(a+b)\). Here, a is \(40^{\circ}\) and b is \(20^{\circ}\).

Step 2: Apply the identity

Substitute the values of a and b into the identity. This simplifies the expression as \(\sin(40^{\circ}+20^{\circ}) = \sin60^{\circ}\).

Step 3: Compute the value

The value of \(\sin60^{\circ}\) is \(\frac{\sqrt{3}}{2}\). Hence, the expression simplifies to \(\frac{\sqrt{3}}{2}\).