Question

How do you find the exact values of `costheta` and `sintheta` when `tantheta=1`?

Answer 1

`sin(theta) = cos(theta) = sqrt(2)/2`

OR

`sin(theta) = cos(theta) = -sqrt(2)/2`

Explanation:

Given: `tan(theta) = 1`

Use the identity:

`tan^2(theta) + 1 = sec^2(theta)`

Substitute `1^2` for `tan^2(theta)`

`1^2 + 1 = sec^2(theta)`

`2 = sec^2(theta)`

Because `sec(theta) = 1/cos(theta)` we can change the above equation to:

`cos^2(theta) = 1/2`

`cos(theta) = +-1/sqrt(2)`

`cos(theta) = +-sqrt(2)/2`

`tan(theta) = sin(theta)/cos(theta) = 1`

`sin(theta) = cos(theta)`

`sin(theta) = +-sqrt(2)/2`

Because we are given nothing to determine whether `theta` is in the first of the third quadrant:

`sin(theta) = cos(theta) = sqrt(2)/2`

OR

`sin(theta) = cos(theta) = -sqrt(2)/2`

Answer 2

`cosθ=sinθ=+-sqrt2/2`

Explanation:

Draw a right isosceles triangle. The angles are 45, 45,90
Tan 45 =1,
Using Pythagoras the sides are in the ratio `1:1:sqrt2`
cos 45=sin45=`1/sqrt2`=`sqrt2/2`
But then we need to look at the angle in the third quadrant to get the negative result.