use the figure to find the exact value of the following trigonometric function. tan(2θ)

Explanation:

Step1: Find (\tan\theta)

In a right - triangle, (\tan\theta=\frac{\text{opposite}}{\text{adjacent}}). Here, (\tan\theta = \frac{3}{4})

Step2: Use double - angle formula for tangent

The double - angle formula for tangent is (\tan(2\theta)=\frac{2\tan\theta}{1 - \tan^{2}\theta}) Substitute (\tan\theta=\frac{3}{4}) into the formula: [ \begin{align*} \tan(2\theta)&=\frac{2\times\frac{3}{4}}{1-\left(\frac{3}{4}\right)^{2}}\ &=\frac{\frac{3}{2}}{1-\frac{9}{16}}\ &=\frac{\frac{3}{2}}{\frac{16 - 9}{16}}\ &=\frac{\frac{3}{2}}{\frac{7}{16}}\ &=\frac{3}{2}\times\frac{16}{7}\ &=\frac{24}{7} \end{align*} ]

Answer:

(\frac{24}{7})