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

2 Answers
Nov 15, 2016

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

Nov 15, 2016

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.