7.3: Double-Angle, Half-Angle, and Product-Sum Formulas
Below are four sets of trigonometric formulas that are useful!
Double-Angle Formulas
\begin{align}
\sin 2x &= 2\sin x \cos x \\
\cos 2x &= \cos^2 x - \sin^2 x \\
&= 1 - 2\sin^2 x \\
&= 2\cos^2 x - 1 \\
\tan 2x &= \dfrac{2\tan x}{1 - \tan^2 x}
\end{align}
These identites are useful for expressions like $\sin 3x$ and $\sin 4x$. We can write
\begin{align}
\sin 3x &= \sin (x + 2x) & \\
&= \sin x \cos 2x + \cos x \sin 2x & \text{Addition identity}
\end{align}
and also \[\sin 4x = \sin\bigg(2(2x)\bigg) = 2\sin 2x \cos 2x\]
Notice how both of the expressions now contain $\sin2x$ and $\cos 2x$. These can be simplified using the double angle formula again.
Prove the identity \[\dfrac{\sin 3x}{\sin x \cos x} = 4\cos x - \sec x\]
Half-Angle Formulas
\begin{align}
\sin \dfrac{u}{2} &= \pm \sqrt{\dfrac{1 - \cos u}{2}} \\
\cos \dfrac{u}{2} &= \pm \sqrt{\dfrac{1 + \cos u}{2}} \\
\tan \dfrac{u}{2} &= \dfrac{1 - \cos u}{\sin u} = \dfrac{\sin u}{1 + \cos u}
\end{align}
where the sign depends on the quadrant where $\dfrac{u}{2}$ lies.
Find the exact value of $\sin 22.5^\circ$.
Find the exact value of $\cos 15^\circ$.
Product-to-Sum Formulas
Below are four formulas which convert factors into terms.
\begin{align}
\sin u \cos v &= \dfrac{1}{2}\left[\sin(u + v) + \sin(u - v)\right] \\
\cos u \sin v &= \dfrac{1}{2}\left[\sin(u + v) - \sin(u - v)\right] \\
\cos u \cos v &= \dfrac{1}{2}\left[\cos(u + v) + \sin(u - v)\right] \\
\sin u \sin v &= \dfrac{1}{2}\left[\cos(u + v) - \sin(u + v)\right] \\
\end{align}
Express $\sin 3x \sin 5x$ as a sum.
Express $\cos 5x \cos 3x$ as a sum.
Sum-to-Product Formulas
Below are four formulas which covnvert terms into factors.
\begin{align}
\sin x + \sin y &= 2\sin\dfrac{x + y}{2}\cos\dfrac{x - y}{2} \\
\sin x - \sin y &= 2\cos\dfrac{x + y}{2}\sin\dfrac{x - y}{2} \\
\cos x + \cos y &= 2\cos\dfrac{x + y}{2}\cos\dfrac{x - y}{2} \\
\cos x - \cos y &= -2\sin\dfrac{x + y}{2}\sin\dfrac{x - y}{2} \\
\end{align}
Express $\sin 7x + \sin 3x$ as a product.
Prove the identity \[\dfrac{\sin 3x - \sin x}{\cos 3x + \cos x} = \tan x\]