How can I find all possible angles for a given sine value?

[physicsdreams]

Homework Statement

This is from a physics problem, but my question is more mathematically oriented.

After working through the problem, I arrive at the last step.

Sin(x)=.967

The question says that there are two possible angles for x.

The Attempt at a Solution

arcsin(.967) gives me 75.2°, which is one of the correct answers.
I am stumped when it comes to finding the second possible angle, which happens to be approximately 105°

I understand that the inverse trig functions are restricted, so how do I find all possible answers for these kinds of questions.

Thank you!

 

[scurty]

Think of the unit circle. $sin(\theta)$ is the y-coordinate on the unit circle. So, $0.967 = sin(\theta) > 0$ in the first and second quadrants. If you draw a triangle inside the unit circle with angle of 75.2 degrees from the x-axis, note where $sin(\theta)$ is. Notice there is another point with that same height in quadrant 2. Simply taking the mirror of this triangle across the y-axis will give you that point. Now calculate the new angle formed from the x-axis!

Spoiler

The angle formed with the trinagle and the Y-AXIS is 14.8. The new angle will be 75.2 + 2*14.8 = 104.8

 

[tiny-tim]

hi physicsdreams!

I understand that the inverse trig functions are restricted

yes, eg arcsinx is defined as having only one value, the "principal value" (between ±90°)

but the solutions to "sinx = 0.967" are arcsin0.967 + n.360° and 180° - arcsin0.967 + n.360° for any whole number n

(look at a graph of sin)

 

[physicsdreams]

Think of the unit circle. $sin(\theta)$ is the y-coordinate on the unit circle. So, $0.967 = sin(\theta) > 0$ in the first and second quadrants. If you draw a triangle inside the unit circle with angle of 75.2 degrees from the x-axis, note where $sin(\theta)$ is. Notice there is another point with that same height in quadrant 2. Simply taking the mirror of this triangle across the y-axis will give you that point. Now calculate the new angle formed from the x-axis!

Spoiler

The angle formed with the trinagle and the Y-AXIS is 14.8. The new angle will be 75.2 + 2*14.8 = 104.8

Alright, I think I can visuallize it now.
Basically, I take 180-(first angle) and I should get my second angle.
Thanks!

P.S. Do I do the same thing for the other trig functions (cos, tan)?

 

[scurty]

Well, it's not always 180-$\theta$, that just happened to be true in this case. For questions like this that involve sine and cosine, just look at where the point in on the unit circle, find a different point on the circle that has the same sign value, and then find $\theta$ for that point.

Tangent is a little bit more tricky. Think about it geometrically in the unit circle. If $tan(\theta)$ is negative, what two quadrants can it possibly be in? If $tan(\theta)$ is positive, what two quadrants can it be in? Then $tan(\theta)$ = 0, undefined should be much simpler.