Question

Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not use a calculator. -135^{\circ}−135

Answer 1

Explanation:

Let's take a look at the given angle 135°

The sketch of the angle which corresponds to
`-(3\pi)/(4)` unit circle and can be seen in the attached image below;

The trigonometric ratios are as follows for an angle θ on the unit circle:

Trigonometric ratio related ratio on coordinate axes

sin θ
`(y)/(1)`

cos θ
`(x)/(1)`

tan θ
`(y)/(x)`

csc θ
`(1)/(y)`

sec θ
`(1)/(x)`

cot θ
`(x)/(y)`

From the sketch of the image attached below;

The six trigonometric ratio for 135° can be expressed as follows:

`sin (-(3\pi)/(4))= (y)/(1)`

`sin (-(3\pi)/(4))=- (√(2))/(2)`

`cos (-(3\pi)/(4))= (x)/(1)`

`cos (-(3\pi)/(4))= -(√(2))/(2)`

`tan (-(3\pi)/(4))= (y)/(x)`

`tan (-(3\pi)/(4))= (-(√(2))/(2))/(-(√(2))/(2))`

`tan (-(3\pi)/(4))= -(√(2))/(2)} * {-(2)/(√(2))`

`tan (-(3\pi)/(4))= 1`

`csc (-(3\pi)/(4))= (1)/(y) \\ \\ csc (-(3\pi)/(4))=(1)/(-(√(2))/(2)) \\ \\ csc=1 * -(2)/(√(2)) \\ \\csc =-√(2)`

`sec (-(3 \pi)/(4))=(1)/(x) \\ \\ sec = (1)/((-(√(2))/(2))) \\ \\ sec = 1 * -(2)/(√(2)) \\ \\ sec = - √(2)`

`cot(-(3 \pi)/(4)) = (x)/(y) \\ \\ cot(-(3 \pi)/(4)) = (-(√(2))/(2) )/(-(√(2))/(2)) \\ \\ cot(-(3 \pi)/(4))= -(√(2))/(2) } * {-(2)/(√(2))} \\ \\ cot (-(3 \pi)/(4)) = 1`