Determine the exact value of the trigonometric function using the provided figure.

sin 2β

Determine the exact value of the expression using the provided data. 

sin 2β where sinβ = 12/37, β lies in quadrant II.

Determine the exact value of the expression using the provided data. 

sin 2β where sinβ = 9/41, β lies in quadrant II.

Determine the exact value of the following expression by writing it as the cosine of a double angle.

cos² 112.5° - sin² 112.5°

Determine the value of the trigonometric expression:

(2 tan 67.5°)/(1 - tan² 67.5°)

Find the values of sin θ and cos θ if cos 2θ = 7/24 and θ terminates in the third quadrant.

Simplify the following trigonometric expression.

tan 46°/[2(1 - tan² 46°)]