Question Type 1: Writing trigonometric functions in different ways Exercises

Simplify $-2\sin(-\theta) + 3\sin(\pi - \theta)$ in terms of $\sin\theta$.

Prove the identity $\sin3\theta = 3\sin\theta - 4\sin^3\theta$.

Simplify $\sin(\pi + \theta)$ in terms of $\sin\theta$.

Express $\cos2\theta$ in terms of $\sin\theta$ only.

Express $\sin(-\theta)$ using only $\sin\theta$.

Write $f(\theta) = 5\sin\theta + 5\cos\theta$ in the form $R\sin(\theta + \alpha)$, where $R>0$ and $0<\alpha<\frac{\pi}{2}$.

Express $g(x) = \sin x + \sqrt{3}\cos x$ in the form $R\cos(x - \alpha)$, where $R>0$ and $0<\alpha<\tfrac{\pi}{2}$.

Simplify $\sin(2\pi - \theta)$ in terms of $\sin\theta$.

Express $h(x) = 2\sin3x - 2\cos3x$ in the form $R\sin(3x - \alpha)$, where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$.

Simplify $-\sin\bigl(\tfrac{\pi}{2}+\theta\bigr)$ in terms of $\cos\theta$.

Simplify $-3\sin(3\pi - \theta)$ in terms of $\sin\theta$.

Express $\cos\theta - \sin\theta$ in the form $\sqrt{2}\,\sin\bigl(\alpha - \theta\bigr)$, stating the value of $\alpha$.