# The trigonometric functions sec(135^@)

Trigonometric Functions
The trigonometric functions $$\displaystyle{\sec{{\left({135}^{\circ}\right)}}}$$

2021-02-01
To find the value of the trigonometric functions, first, convert sec function in terms of $$\displaystyle{\cos{{\left(\theta\right)}}}$$.
Use the identity $$\displaystyle{\sec{{\left(\theta\right)}}}=\frac{{1}}{{{\cos{{\left(\theta\right)}}}}}$$
We have
$$\displaystyle\theta={135}^{\circ}={90}^{\circ}+{45}^{\circ}$$
$$\displaystyle{\cos{{\left({135}^{\circ}\right)}}}={\cos{{\left({90}^{\circ}+{45}^{\circ}\right)}}}$$
$$\displaystyle=-{\sin{{\left({45}^{\circ}\right)}}}{\left\lbrace{\cos{{\left({90}^{\circ}+\theta\right)}}}=-{\sin{{\left(\theta\right)}}}\right.}$$
$$\displaystyle=-\frac{{1}}{\sqrt{{2}}}$$
Now $$\displaystyle{\sec{{\left(\theta\right)}}}=\frac{{1}}{{{\cos{{\left(\theta\right)}}}}}$$
$$\displaystyle{\sec{{\left({135}^{\circ}\right)}}}=\frac{{1}}{{{\cos{{\left({135}^{\circ}\right)}}}}}$$
$$\displaystyle=\frac{{1}}{{-{\sin{{\left({45}^{\circ}\right)}}}}}$$
$$\displaystyle=\frac{{1}}{{-\frac{{1}}{\sqrt{{2}}}}}$$
$$\displaystyle=-\sqrt{{2}}$$
=-1.4142