Find an angle θ with 0 < θ < 2π radians that has the same:

Question

Sine function value has 7π/6

θ= ____ radians

Cosine function value as 7π/6

θ= ____ radians

Answer 1

For anything like this, start by drawing a picture.

7pi / 6 is just a bit larger than pi. Thus the angle is in the 3rd quadrant, and the angle from the -x axis is pi/6.

The sine of pi/6 is 0.5, so the sine of 7pi / 6 will also be - 0.5 (Confirm that this makes sense by noting that the opposite side of the angle in the 3rd quadrant is negative)

To check, we can calculate the sine of 7pi / 6. This also gives us - 0.5

Use the same approach for the cosine

Answer 2

Think about your unit circle here. 7π/6 is just 1/6 of a π past π. (30º past 180º if you prefer that.)

Now, the cosine and sine of an angle are just the x and y coordinates of that point on the unit circle. Looking at 7π/6, you can see that it shares an x coordinate with 5π/6, so cos 7π/6 = cos 5π/6. (You can also check your unit circle or use 30-60-90 triangle measurements to figure out that that cosine value is -√(3)/2 .)

Now, the y-value: sin 7π/6 = sin 11π/6 because they're both π/6 below the x-axis. Also, sin 7π/6 = -1/2.

If you're not super comfortable with drawing a unit circle on your own and filling in all the values, and why they're there, either find a way to learn that or get in touch, and I'll help. If you really "get" the unit circle, you can ace trig!

Have fun!