Question

What is the Trigonometric Value of cos θ = -2/3, 90º < θ < 180º; sin θ?

Answer 1

`sintheta=sqrt5/3`

Explanation:

`"using the "color(blue)"trigonometric identity"`

`•color(white)(x)sin^2theta+cos^2theta=1`

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

`"since "90^@ < theta<180^@`

`"then "theta" in second quadrant where "sintheta>0`

`sintheta=sqrt(1-(-2/3)^2)`

`color(white)(sintheta)=sqrt(5/9)=sqrt5/3`

Answer 2

See explanation.

Explanation:

The task is to calculate `sin theta` when `cos theta` is given. To do this we can use the following identity:

`sin^2theta+cos^2theta=1`

If we substitute the given data we get:

`sin^2theta+(-2/3)^2=1`

`sin^2theta+4/9=1`

`sin^2theta=1-4/9`

`sin^2theta=5/9`

This equation has 2 solutions:

`sin theta=-sqrt(5)/3 vv sin theta=sqrt(5)/3`

To decide which value is correct we have to use the information given. If the angle is in the 2nd quadrant (`90^o < theta < 180^o`) then `sin theta` is positive, so we have to choose:

`sin theta=sqrt(5)/3`