# Coordinates where gradient is zero of trig functions

#### DomB

Hi everyone, I have two questions I am struggling with. They are: work out the coordinates when the gradient of these functions is equal to zero
t=cos(theta)
and
Z=sin(2theta)

I think I need to find the derivatives, but then what?
have a nice day.

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#### skeeter

Math Team
Hi everyone, I have two questions I am struggling with. They are: work out the coordinates when the gradient of these functions is equal to zero
t=cos(theta)
and
Z=sin(2theta)

I think I need to find the derivatives, but then what?
Set each derivative equal to zero and solve for $\theta$.

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• 1 person

#### DomB

Think I just get 0,0 by doing that.

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#### skeeter

Math Team
Think I just get 0,0 by doing that.
$t =\cos{\theta}$

$\dfrac{dt}{d\theta} = -\sin{\theta} = 0 \implies \theta = k\pi \, , \, k \in \mathbb{Z}$

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$z = \sin(2\theta)$

$\dfrac{dz}{d\theta} = 2\cos(2\theta) = 2(2\cos^2{\theta} - 1) = 0 \implies \cos{\theta} = \pm \dfrac{1}{\sqrt2}$

so, $\theta = \, ?$

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#### skipjack

Forum Staff
$\cos(2\theta) = 0$ directly implies that $2\theta = \pi/2 + k\pi$.

#### Country Boy

Math Team
Think I just get 0,0 by doing that.
Are you just finding arcsin(0) on your calculator? You need to know more about sin(x) than that to do these problems!

sin(x)= 0 for x any multiple of $$\displaystyle \pi$$.

cos(x)= 0 for x any odd multiple of $$\displaystyle \frac{\pi}{2}$$.