Prove that \displaystyle \frac{1-2\sin^2x}{\cos x + \sin x} + 2\sin \frac{x}{2}\cos \frac{x}{2} = \cos x .

Solve for x. \displaystyle \cos 2x + 2\sin^2x - 3 =-2 , where 0\leq x \leq 2\pi

Determine the solution(s) for the of the following equation, where 0\leq x \leq 2\pi.

\displaystyle \cos x = \frac{\sqrt{3}}{2}

Determine the solution(s) for the of the following equation, where 0\leq x \leq 2\pi.

\displaystyle \tan x = - \sqrt{3}

Determine the solution(s) for the of the following equation, where 0\leq x \leq 2\pi.

\displaystyle \sin x = - \frac{\sqrt{2}}{2}

The quadratic trigonometric equation a \cos2 x +b\cos x - 1 = 0 has the solutions \frac{\pi}{3}, \pi, and \frac{5\pi}{3} in the interval 0 \leq x \leq 2\pi. What are the values of a and b?

The depth of the ocean at a swim buoy can be modelled by the function d(t) = 4 + 2\sin(\frac{\pi}{6}t), where d is the depth of water in metres and t is the time in hours, if 0 \leq t \leq 24. Consider a day when t = 0 represents midnight. Determine when the depth of water is 3 m.

Nina needs to find the cosine of \frac{11\pi}{4}. If she knows the sine and cosine of \pi, as well as the sine and cosine of \frac{7\pi}{4}, how can she find the cosine of \frac{11\pi}{4}? What is her answer?

Solve 3 \sin x + 2 = 1.5, where 0\leq x \leq 2\pi.

The tangent of the acute angle a is 0.75, and the tangent of the acute angle \beta is 2.4. Without using a calculator, determine the value of \sin (\alpha - \beta) and \cos(\alpha + \beta).

The angle x lies in the interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{4}{9}. Determine the value of the following. State your answer in exact values.

\displaystyle \sin 2x

The angle x lies in the interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{4}{9}. Determine the value of the following. State your answer in exact values.

\displaystyle \cos 2x

The angle x lies in the interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{4}{9}. Determine the value of the following. State your answer in exact values.

\displaystyle \cos \frac{x}{2}

The angle x lies in the interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{4}{9}. Determine the value of the following. State your answer in exact values.

\displaystyle \sin 3x

Solve for x when -2\pi \leq x \leq 2\pi.

\displaystyle 2-14\cos x = -5

Solve for x when -2\pi \leq x \leq 2\pi.

\displaystyle 9-22\cos x - 1= 19

Solve for x when -2\pi \leq x \leq 2\pi.

\displaystyle 2 + 7.5 \cos x = -5.5