Question

Given tan theta = -3/2, theta in Quadrant IV find sin 2theta , cos 2theta, tan 2 theta

Answer 1

To clarify, you have to find \(\sin2\theta\) and so on, not \(\sin^2\theta\)?

Answer 2

Yes

Answer 3

I do know that in order to start I need to find r. using the pythagorean theorem.

Answer 4

2 is x and -3 is y?

Answer 5

Okay, there are a few ways to go about this, but I think the easiest is to use your identities. I'm not sure if your method is the same or not...

Recall that \(\tan ax=\dfrac{\sin ax}{\cos ax}\), so finding tangent is a matter a finding sine and cosine.

Given that \(\tan\theta=-\dfrac{3}{2}\), you have
\[\frac{\sin\theta}{\cos\theta}=-\dfrac{3}{2}\]
Now you can say \(\sin\theta=-3\) and \(\cos\theta=2\), or you can say \(\sin\theta=3\) and \(\cos \theta=-2\). To know for sure which one is the case, you're given that the angle lies in QIV. Any angle in the fourth quadrant has positive values of cosine and negative values of sine, so
\[\frac{\sin\theta}{\cos\theta}=-\dfrac{3}{2}~~\Rightarrow~~\sin\theta=-3,~\cos\theta=2\]

Answer 6

OHHHH that is definitely making more sense to me.

Answer 7

Anyway, recall another identity: \(\sin2\theta=2\sin\theta\cos\theta\). You know the exact values of sine and cosine, so \(\sin2\theta=2(-3)(2)=-12\).

Another identity: \(\cos2\theta=\cos^2\theta-\sin^2\theta\). Then you have \(\cos2\theta=2^2-(-3)^2=-5\).

Finally, \(\tan2\theta=\dfrac{\sin2\theta}{\cos2\theta}=\dfrac{-12}{-5}=\dfrac{12}{5}\).

Answer 8

what about the radius?

Answer 9

we are suppose to use r=sqrt x^2+y^2

Answer 10

I wasn't aware that you had to find a radius, but you certainly could. \(y=12\) and \(x=5\).

Answer 11

question. I am working on a problem. and it states tan theta =-15/8 . the solution is that x is 8 and y is -15. Why is this? I thought that it sin theta/cos theta? therefore x should be 15 and y should be -8. But this is incorrect. please expalin.

Answer 12

Wow. I am an idiot. You are awesome.

Answer 13

:)

Answer 14

I have been sitting here for hours. I swear. I am learning pre calc online. Please don't log off. lol

Answer 15

I come and go throughout the day. If I'm not around, there are plenty of other people that are capable.

Answer 16

I am finishing up my question now.

Answer 17

so alternatively, if I have tan theta = -7/2 y = -7 and x =2

Answer 18

because sine theta is negative and cosine theta is positive