Suppose that sin q = − 12/13 and that q is a Quadrant IV angle.
(a) Find the exact value of cos q. Show work.
(b) Find the exact value of sin 2q. Show work.
(a) To find the exact value of cos q, we can use the Pythagorean identity: sin^2 q + cos^2 q = 1. Since sin q = -12/13, we can substitute this value into the equation:
(-12/13)^2 + cos^2 q = 1
Simplifying:
144/169 + cos^2 q = 1
cos^2 q = 169/169 - 144/169
cos^2 q = 25/169
Since q is a Quadrant IV angle, cos q is positive. Taking the square root of both sides:
cos q = √(25/169) = 5/13
Therefore, the exact value of cos q is 5/13.
(b) To find the exact value of sin 2q, we can use the double-angle formula for sine: sin 2q = 2sin q cos q. We already know the value of sin q as -12/13 and we just found the value of cos q as 5/13.
Substituting these values into the formula:
sin 2q = 2(-12/13)(5/13)
sin 2q = -120/169
Therefore, the exact value of sin 2q is -120/169.