# Thread: Given tan (theta) what is sin (theta)

1. ## Given tan (theta) what is sin (theta)

if pi/2 ≤ theta ≤ 3pi/2 and tanθ = -4/3
Determine the exact value of sinθ

if the answer was something I could use the special triangles to solve It wouldnt be so hard ie(1/2, root3/2 ....)

I could maybe draw it out and solve, but how do you do something like this?

2. Originally Posted by brentwoodbc
if pi/2 ≤ theta ≤ 3pi/2 and tanθ = -4/3
Determine the exact value of sinθ

if the answer was something I could use the special triangles to solve It wouldnt be so hard ie(1/2, root3/2 ....)

I could maybe draw it out and solve, but how do you do something like this?
$\displaystyle \frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}$ means that you're in either the 2nd or 3rd quadrant. But tan is negative in the 3rd quadrant and the 4th quadrant. Therefore $\displaystyle \theta$ lies in the second quadrant.

Now draw a right-triangle with angle $\displaystyle \theta$ such that the opposite side is 4 and the adjacent side is 3. Clearly the hypotenuse is 5. Then $\displaystyle \sin \theta = \frac{4}{5}$ (because sine is positive in the 2nd quadrant).

3. how do you know what quadrant it is in? how do you get theta from the triangle?

4. Originally Posted by brentwoodbc
Thank you.

+ or - ? answers given are

-4/5

-3/5

3/5

4/5
I edited my original post because I made a careless mistake.

Originally Posted by brentwoodbc
[snip] how do you get theta from the triangle?
Use basic trigonometry to get the magnitude of $\displaystyle \sin \theta$. Then decide whether it's plus or minus.

5. Im not allowed a calc. so how else would I find if it's plus/ neg.

6. Originally Posted by brentwoodbc
[snip]
Im not allowed a calc. so how else would I find if it's plus/ neg.
I assume that you're expected to be familiar with how the definition of sine (and cosine) relates to the unit circle. That's why I have refered to quadrants. You're expected to know what the sign of sine (and cosine) and tan is in each quadrant.

7. Originally Posted by mr fantastic
$\displaystyle \frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2}$ means that you're in either the 2nd or 3rd quadrant. But tan is negative in the 3rd quadrant and the 4th quadrant. Therefore $\displaystyle \theta$ lies in the second quadrant.

Now draw a right-triangle with angle $\displaystyle \theta$ such that the opposite side is 4 and the adjacent side is 3. Clearly the hypotenuse is 5. Then $\displaystyle \sin \theta = \frac{4}{5}$ (because sine is positive in the 2nd quadrant).
thanks that makes more sense, but isnt tan supposed to be negative?
(tan(theta)=-4/3)

8. Originally Posted by brentwoodbc
thanks that makes more sense, but isnt tan supposed to be negative?
(tan(theta)=-4/3)
In the second quadrant tan is negative and sine is positive. The triangle I used earlier gives you the size of $\displaystyle \sin \theta$. Then you examine the quadrant the angle is in to decide whether the answer is +ve or -ve.