Question

Can someone help me answer this question? 'find the exact value of Cos theta, given sin theta = 5/9 and cos theta < 0

Answer 1

I got `0.831`

Explanation:

Have a look:

Answer 2

`cos(theta)= -0.83147941928309808568527749607034`

= -0.8315 (4dp)

Explanation:

Find `theta` (Note: there are 2 possible angles for `sin(theta) = 5/9`)
`Sin^-1(5/9) = 33.748988595888587817404332744458 (degrees)`

The sin will also have the value `(5/9)` at

`180-33.748988595888587817404332744458`

`=146.25101140411141218259566725554 (degrees)`

If you are not sure which of these angles will have cos < 1, try both.
(note that the numbers for both angles are actually the same but one is positive and the other is negative)

`cos(33.748988595888587817404332744458)`
`=0.83147941928309808568527749607034`

`cos(146.25101140411141218259566725554)`
`=-0.83147941928309808568527749607034`

(so `theta` is the 146.25101.... value)

Answer 3

`costheta=-sqrt56/9`

Explanation:

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

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

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

`"given "sintheta=5/9" and "costheta<0`

`"then "theta" is in the second quadrant"`

`costheta=-sqrt(1-(5/9)^2)`

`color(white)(costheta)=-sqrt(1-25/81)`

`color(white)(costheta)=-sqrt(56/81)=-sqrt56/9`