Question

Given `costheta=-4/(sqrt20)`, where `0^@ <= theta <= 180^@` ?

Answer 1

`theta ~~ 153.4^@`

Explanation:

Given `cos(theta)=-4/(sqrt20), 0^@ <= theta <= 180^@`

Use the inverse cosine on both sides:

`theta = cos^-1(-4/(sqrt20)), 0^@ <= theta <= 180^@`

`theta ~~ 153.4^@`

You have the value for the cosine function the denominator should be rationalized:

`cos(theta)= -4/sqrt20`

`cos(theta)= -4/(2sqrt5)`

`cos(theta)= (-2sqrt5)/5`

The secant function is the reciprocal of the cosine function:

`sec(theta) = 1/cos(theta)`

`sec(theta) = -sqrt5/2`

The sine function can be found using the identity:

`sin(theta) = +-sqrt(1-cos^2(theta))`

`sin(theta) = +-sqrt(1-((-2sqrt5)/5)^2)`

`sin(theta) = +-sqrt(25/25- 20/25)`

`sin(theta) = +-sqrt5/5`

We know that the sine function is positive in the second quadrant:

`sin(theta) = sqrt5/5`

The cosecant function is the reciprocal of the sine function:

`csc(theta) = 1/sin(theta)`

`csc(theta) = 5/sqrt5`

`csc(theta) = sqrt5`

Find the tangent function using the identity:

`tan(theta) = sin(theta)/cos(theta)`

`tan(theta) = (sqrt5/5)/((-2sqrt5)/5)`

`tan(theta) = -1/2`

The cotangent function is the reciprocal of the tangent function:

`cot(theta) = -2`