Problem 28 - Precalculus Essentials

Question

Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$\frac{\tan 50^{\circ}-\tan 20^{\circ}}{1+\tan 50^{\circ} \tan 20^{\circ}}$$

Step-by-Step Solution

Answer

The exact value of the expression is \(\frac{\sqrt{3}}{3}\).

Step 1: Identify the Trigonometric Identity

The given expression aligns with the trigonometric identity \(\frac{\tan A - \tan B}{1 + \tan A \tan B} = \tan (A - B)\). Here, A is 50 and B is 20.

Step 2: Apply the Trigonometric Identity

Apply the aforementioned identity, replace A and B respectively with 50 and 20, in order to simplify the overall expression. The result will be \(\tan (50 - 20) = \tan 30^\circ\) .

Step 3: Find the Tangent Value

Having simplified the expression down to the tangent of an angle, the next step is to calculate the value of \(\tan 30^\circ\). Using a trigonometric values table or calculator, we find \(\tan 30^\circ\) to be \(\frac{\sqrt{3}}{3}\).