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Trigonometric equations
Trigonometric identities and equations
CatalogueDPMathematicsMathematics: Analysis and Approaches HLGeometry and trigonometry

Trigonometric identities and equations

Select useful identities, solve on a stated interval, and retain every valid trigonometric solution.

  • AA HL
  • Trigonometric equations
  • Papers 1 and 2
  • GDC optional
  • About 25 min

By the end of this lesson

  • use Pythagorean, compound-angle and double-angle identities
  • transform equations into a solvable form
  • generate and filter all solutions on an interval

Why this matters. The main difficulty is usually not algebra alone. It is choosing a form that exposes the equation without losing or inventing solutions.

Choose an identity with a purpose

$\sin^2x+\cos^2x=1$$\sin2x=2\sin x\cos x$$\cos2x=1-2\sin^2x=2\cos^2x-1$$\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B$

Choose the version that reduces the number of different functions or angles in the equation.

Checkpoint

To solve $2\cos^2x-3\sin x=0$, which substitution is most direct?
Show explanation

It turns the equation into a quadratic in $\sin x$.

Example 1

Solve a quadratic form

Keep only valid sine values

Solve $2\cos^2x-3\sin x=0$ for $0\le x<2\pi$.

$$2(1-\sin^2x)-3\sin x=0$$
$$(2\sin x-1)(\sin x+2)=0.$$

$\sin x=-2$ is impossible. From $\sin x=\frac12$, the interval gives

$$x=\frac\pi6,\ \frac{5\pi}{6}.$$

Protect solutions during algebra

Dividing by $\sin x$ or $\cos x$ can discard cases where that factor is zero. Factor first and solve each branch.

Checkpoint

For $\sin x(2\cos x-1)=0$, what must be solved?
Show explanation

A product is zero when either factor is zero.

Example 2

Use a compound angle

Exact evaluation

Find $\sin75^\circ$ exactly.

$$\sin(45^\circ+30^\circ)=\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ$$
$$=\frac{\sqrt2}{2}\frac{\sqrt3}{2}+\frac{\sqrt2}{2}\frac12=\frac{\sqrt6+\sqrt2}{4}.$$

The signs follow directly from the addition identity.

Checkpoint

If $\cos(2x)=0$ and $0\le x<\pi$, which set is complete?
Show explanation

$0\le2x<2\pi$, so $2x=\pi/2,3\pi/2$.

Example 3

Combine into one sinusoid

Solve $\sin x+\cos x=1$

Since $\sin x+\cos x=\sqrt2\sin(x+\pi/4)$, solve

$$\sin\left(x+\frac\pi4\right)=\frac1{\sqrt2}.$$

For $0\le x<2\pi$, this gives $x+\pi/4=\pi/4,3\pi/4$ modulo $2\pi$, hence

$$x=0,\ \frac\pi2.$$

Substitution confirms both values.

In the exam

Quick check

Five questions to check the main decisions from this lesson.

Your score is not saved.

  1. $1-\sin^2x$ equals…
    Show explanation

    Rearrange the Pythagorean identity.

  2. $2\sin x\cos x$ equals…
    Show explanation

    This is the double-angle identity for sine.

  3. Why is dividing by $\cos x$ risky?
    Show explanation

    Division assumes the divisor is non-zero.

  4. On $0\le x<2\pi$, $\sin x=0$ at…
    Show explanation

    $2\pi$ is excluded by the half-open interval.

  5. If $\cos2x=1$, one valid family is…
    Show explanation

    $2x=2k\pi$, so $x=k\pi$.

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