Question

find the exact values of the six trigonometic functions for the following angle. 135°
sin 135° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
cos 135° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
tan 135° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
cot 135° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
sec 135° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
csc 135° =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Recall angle - relation

$135^{\circ}=180^{\circ} - 45^{\circ}$, and in the second - quadrant, $\sin$ is positive, $\cos$ is negative.

Step2: Find $\sin135^{\circ}$

$\sin135^{\circ}=\sin(180^{\circ}-45^{\circ})=\sin45^{\circ}=\frac{\sqrt{2}}{2}$

Step3: Find $\cos135^{\circ}$

$\cos135^{\circ}=\cos(180^{\circ}-45^{\circ})=-\cos45^{\circ}=-\frac{\sqrt{2}}{2}$

Step4: Find $\tan135^{\circ}$

$\tan135^{\circ}=\frac{\sin135^{\circ}}{\cos135^{\circ}}=\frac{\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}}=- 1$

Step5: Find $\cot135^{\circ}$

$\cot135^{\circ}=\frac{1}{\tan135^{\circ}}=-1$

Step6: Find $\sec135^{\circ}$

$\sec135^{\circ}=\frac{1}{\cos135^{\circ}}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}$

Step7: Find $\csc135^{\circ}$

$\csc135^{\circ}=\frac{1}{\sin135^{\circ}}=\frac{1}{\frac{\sqrt{2}}{2}}=\sqrt{2}$

Answer:

$\sin135^{\circ}=\frac{\sqrt{2}}{2}$
$\cos135^{\circ}=-\frac{\sqrt{2}}{2}$
$\tan135^{\circ}=-1$
$\cot135^{\circ}=-1$
$\sec135^{\circ}=-\sqrt{2}$
$\csc135^{\circ}=\sqrt{2}$