Using Identities to Solve Equations

When One Function Isn’t Enough

The basic method works when an equation has a single trig function. But real equations are messier:

$2\cos^2 x + \sin x = 1$

This has two different functions and a square. You cannot isolate the angle as it stands.

The fix is always the same idea: use an identity to rewrite the equation in terms of one function, then solve it like before.

Move everything to one side and the equation becomes “where does this curve hit zero?” The green dots are the answers. Our job is to find them with algebra instead of by looking.

Strategy 1: Trade with the Pythagorean Identity

The square of one function can be swapped for the other using $\sin^2 x + \cos^2 x = 1$.

Start by moving everything to one side, then replace $\cos^2 x$ with $1 - \sin^2 x$:

Now it is a quadratic in $\sin x$. Factor it:

$(2\sin x + 1)(\sin x - 1) = 0$

So $\sin x = -\frac{1}{2}$ or $\sin x = 1$. Each is now a basic equation:

• $\sin x = -\tfrac{1}{2} \;\Rightarrow\; x = 210°, 330°$
• $\sin x = 1 \;\Rightarrow\; x = 90°$

Those are exactly the three zeros in the animation.

Strategy 2: Factor, Never Divide

Some equations share a common factor. Pull it out, but do not divide it away.

$\sin 2x = \sin x$

Use the double angle identity $\sin 2x = 2\sin x\cos x$, move everything over, and factor:

Set each factor to zero: $\sin x = 0$ or $\cos x = \frac{1}{2}$.

Never divide both sides by $\sin x$. If you do, you throw away every solution where $\sin x = 0$. Factor and set each piece to zero instead.

Strategy 3: Reduce with a Double Angle

When an equation mixes $2x$ and $x$, a double angle identity brings them together.

$\cos 2x + \cos x = 0$

Replace $\cos 2x$ with $2\cos^2 x - 1$:

So $\cos x = \frac{1}{2}$ or $\cos x = -1$, both basic equations.

The Recipe

Every one of these follows the same four moves:

Identities are the bridge: they turn a tangled trig equation into ordinary algebra you already know how to finish.