# Use the figures to find the exact value of the trigonometric function tan2 theta. 01510200841.png

Question
Trigonometric Functions
Use the figures to find the exact value of the trigonometric function $$\displaystyle{\tan{{2}}}\theta$$.

2021-03-03
Using the triangle ,
$$\displaystyle{\tan{\theta}}=\frac{{3}}{{4}}$$
Apply the formula of the trigonometric function ,
$$\displaystyle{\tan{{2}}}\theta={\left({2}{\tan{\theta}}\right)}{\left({1}−{\tan{{2}}}\theta\right)}$$
Substitute the value of $$\displaystyle{\tan{\theta}}=\frac{{3}}{{4}}$$ in the formula,
$$\displaystyle{\tan{{2}}}\theta=\frac{{{2}\cdot{\left(\frac{{3}}{{4}}\right)}}}{{{1}-{\left(\frac{{3}}{{4}}\right)}^{{2}}}}$$
Solving the equation above,
$$\displaystyle{\tan{{2}}}\theta=\frac{{\frac{{3}}{{2}}}}{{{1}-\frac{{9}}{{16}}}}$$
$$\displaystyle=\frac{{\frac{{3}}{{2}}}}{{\frac{{7}}{{16}}}}$$
$$\displaystyle=\frac{{24}}{{7}}$$
Thus the value of the trigonometric function $$\displaystyle{\tan{{2}}}\theta{i}{s}\frac{{24}}{{7}}.$$

### Relevant Questions

$$\sec \theta = -3, \tan \theta > 0$$. Find the exact value of the remaining trigonometric functions of
$$\theta$$.

Find the exact value of each of the remaining trigonometric function of $$\theta.$$
$$\displaystyle \cos{\theta}=\frac{24}{{25}},{270}^{\circ}<\theta<{360}^{\circ}$$
$$\displaystyle \sin{\theta}=?$$
$$\displaystyle \tan{\theta}=?$$
$$\displaystyle \sec{\theta}=?$$
$$\displaystyle \csc{\theta}=?$$
$$\displaystyle \cot{\theta}=?$$

Sketch a right triangle corresponding to the trigonometric function of the acute angle theta. Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of theta. $$\displaystyle{\cos{\theta}}=\frac{{21}}{{5}}$$
Find the exact value of the trigonometric function $$\displaystyle{\sec{{\left(\frac{{-{9}\pi}}{{4}}\right)}}}$$ .
Find the exact value of the trigonometric function $$\displaystyle\frac{{\cos{{\left({9}\pi\right)}}}}{{4}}$$.
Use a calculator to find the value of the trigonometric function $$\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{10}}\right)}}}$$ to four decimal places.
$$\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{4}}\right)}}}$$
$$\displaystyle{{\sec{{6.7}}}^{\circ},}{\cos{{e}}}{c}{83.3}^{\circ}$$
Find the values of the other trigonometric functions of theta if $$\displaystyle{\cot{\theta}}=-\frac{{4}}{{3}}{\quad\text{and}\quad}{\sin{\theta}}{<}{0}$$.