Trigonometry

If \( \sin \theta = \frac{3}{5} \) and \( \theta \) is an acute angle, what is \( \cos \theta \)?

In a right triangle, if the angle \( A = 30^\circ \), what is the ratio of the length of the side opposite to \( A \) to the hypotenuse?

Find \( \tan 45^\circ \).

If \( \cos \theta = \frac{5}{13} \) and \( \theta \) is in the first quadrant, find \( \sin \theta \).

Evaluate \( \sin 60^\circ \cos 30^\circ + \cos 60^\circ \sin 30^\circ \).

If \( \tan \theta = 3 \) and \( \theta \) is acute, find \( \sin \theta \).

Prove that \( 1 + \tan^2 \theta = \sec^2 \theta \).

If \( \sin A = \frac{5}{13} \) and \( \cos B = \frac{12}{13} \), find \( \sin (A + B) \) assuming both angles are acute.

Solve for \( \theta \) if \( 2 \sin^2 \theta - 3 \sin \theta + 1 = 0 \) and \( 0^\circ \leq \theta \leq 180^\circ \).

In triangle \( ABC \), if \( \sin A = \frac{3}{5} \), \( \sin B = \frac{5}{13} \), find \( \sin C \).