# Find the exact value of the trigonometric function sec ((-9pi)/4) .

Question
Trigonometric Functions
Find the exact value of the trigonometric function $$\displaystyle{\sec{{\left(\frac{{-{9}\pi}}{{4}}\right)}}}$$ .

2020-11-27
We know that
$$\displaystyle{\sec{{\left(-{x}\right)}}}={\sec{{\left({x}\right)}}}$$
Therefore,
$$\displaystyle{\sec{{\left(\frac{{-{9}\pi}}{{4}}\right)}}}={\sec{{\left(\frac{{{9}\pi}}{{4}}\right)}}}$$
Rewriting the above trigonometric function,
$$\displaystyle{\sec{{\left(\frac{{{9}\pi}}{{4}}\right)}}}={\sec{{\left(\frac{{{8}\pi+\pi}}{{4}}\right)}}}$$
$$\displaystyle={\sec{{\left(\frac{{{8}\pi}}{{4}}+\frac{\pi}{{4}}\right)}}}$$
$$\displaystyle={\sec{{\left({2}\pi+\frac{\pi}{{4}}\right)}}}$$
We know that $$\displaystyle{2}\pi+\theta$$ lies in first quadrant and in first quadrant all trigonometric functions are positive.
Therefore,
$$\displaystyle{\sec{{\left({2}\pi+\frac{\pi}{{4}}\right)}}}={\sec{{\left(\frac{\pi}{{4}}\right)}}}$$
$$\displaystyle=\sqrt{{2}}$$
Hence, exact value of given trigonometric function is $$\displaystyle\sqrt{{2}}$$

### Relevant Questions

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

Find the exact value of the trigonometric function $$\displaystyle\frac{{\cos{{\left({9}\pi\right)}}}}{{4}}$$.
The question asks for the exact value of the trigonometric function at the given real number:
$$\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{4}}\right)}}}$$
Use the figures to find the exact value of the trigonometric function $$\displaystyle{\tan{{2}}}\theta$$.
$$\displaystyle{{\sec{{6.7}}}^{\circ},}{\cos{{e}}}{c}{83.3}^{\circ}$$
Use a calculator to find the value of the trigonometric function $$\displaystyle{\sin{{\left(\frac{{{3}\pi}}{{10}}\right)}}}$$ to four decimal places.
The trigonometric functions $$\displaystyle{\sec{{\left({135}^{\circ}\right)}}}$$
The trigonometric functions $$\displaystyle{\sec{{\left({225}^{\circ}\right)}}}$$