Unipal Initiative - Free Competition Math Courses, From Students For Students

Angle-sum and difference formulas

\sin(A\pm B) = \sin A\cos B \pm \cos A \sin B

\cos(A\pm B) = \cos A\cos B \mp \sin A \sin B

Double-angle formulas

\sin(2\theta) = 2\sin\theta\cos\theta

\cos(2\theta) = \cos^2\theta-\sin^2\theta = 2\cos^2\theta-1 = 1-2\sin^2\theta

The three equivalent forms of $\cos(2\theta)$ come from substituting $\sin^2\theta+\cos^2\theta=1$ — pick whichever form matches the rest of your expression.

Half-angle formulas

\sin\left(\dfrac\theta2\right) = \pm\sqrt{\dfrac{1-\cos\theta}{2}}, \qquad \cos\left(\dfrac\theta2\right) = \pm\sqrt{\dfrac{1+\cos\theta}{2}}

The sign depends on which quadrant $\theta/2$ falls in — always check this before finalizing an answer.

Solving trig equations

The general method: use identities to reduce the equation to a single trig function, factor or apply the quadratic formula if needed, solve for that function’s value, then find every angle in the given interval with that value — remembering trig functions are periodic, so there’s often more than one solution.

Worked example

Problem: Solve $\sin(2x) = \cos(x)$ for $x \in [0, 2\pi)$ .

Solution: Rewrite $\sin(2x) = 2\sin x\cos x$ :

2\sin x\cos x = \cos x \;\Longrightarrow\; \cos x(2\sin x - 1) = 0

So either $\cos x = 0$ or $\sin x = \tfrac12$ .

$\cos x = 0$ : $x = \tfrac\pi2, \tfrac{3\pi}2$ .

$\sin x = \tfrac12$ : $x = \tfrac\pi6, \tfrac{5\pi}6$ .

All four solutions: $x \in \left\{\tfrac\pi6, \tfrac\pi2, \tfrac{5\pi}6, \tfrac{3\pi}2\right\}$ .

Practice problems

1.Find the exact value of $\cos(15°)$ using a difference formula.

2.If $\sin\theta = \tfrac35$ with $\theta$ in Quadrant I, find $\sin(2\theta)$ .

3.Solve $2\cos^2x - 1 = 0$ for $x \in [0, 2\pi)$ .

4.If $\cos\theta = \tfrac{7}{25}$ with $\theta$ in Quadrant I, find $\cos(2\theta)$ .

Trig Identities & Equations

Angle-sum and difference formulas

Double-angle formulas

Half-angle formulas

Solving trig equations

Practice problems