Proving Non-Negative Numbers in a^2+b^2=1

[armolinasf]

Homework Statement

We that if there are two non negative numbers, a and b, such that a^2+b^2=1 then there exists an angle theta such that sin(theta)=a and cos(theta)=b. If I wanted to show that this is true even for negative numbers, would it be enough to say that if either a or b were negative it wouldn't matter since it would removed when we square the numbers in a^2+b^2=1 and the relationship would hold true? Thanks for the help.

 

[Mark44]

Homework Statement

We that if there are two non negative numbers, a and b, such that a^2+b^2=1 then there exists an angle theta such that sin(theta)=a and cos(theta)=b. If I wanted to show that this is true even for negative numbers, would it be enough to say that if either a or b were negative it wouldn't matter since it would removed when we square the numbers in a^2+b^2=1 and the relationship would hold true? Thanks for the help.

Look at this in terms of the unit circle, x² + y² = 1. If x and y are nonnegative, we're working with the upper right quadrant of this circle, so 0 <= θ <= π/2. For any point (a, b) on this quadrant of the circle, cos(θ) = a and sin(θ) = b.

Now look at things on the entire circle to see how it works with a or b being negative.