# Thread: Cos theta = -Sin theta

1. ## Cos theta = -Sin theta

For what angles theta B/T 0 & 360 does cos theta = -sin theta?

It's simple question, but I could not think of any angles that would make this statement true. Maybe I'm missing something...

2. Here's a hint -- there are two places between 0 and 360 degrees where cos(theta) = -sin(theta). Can you see that at 45 degrees cos(theta) = sin(theta)? For this angle you are in quadrant 1, where both sin and cos are positive. But what about other quadrants where sin and cos have opposiite signs? I trust this helps - now post back with what you think the answer is.

3. Yes. They would have opposite sings in Q1 and Q4, when dealing with 45 reference.

4. $\displaystyle Cos\theta$ in a unit circle gives the x co-ordinate of a point on the circumference.

$\displaystyle Sin\theta$ gives the y co-ordinate of the same point.

$\displaystyle -Sin\theta$ is the negative of the y co-ordinate.

To be equal numerically, the angles cut the quadrants in two equal parts,
hence the angles are 45, 135, 225 and 315 degrees.

For 2 of these, the x and y co-ordinates of the 4 points are equal numerically and have opposite sign.

In which quadrants are the co-ordinates of opposite sign?

5. $\displaystyle cos(\theta)=-sin(\theta)$
$\displaystyle cot(\theta)=-1$
$\displaystyle tan(\theta)=-1$

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### costheta - sintheta

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