Solving trigonometric equations

Solving trig equations in radians over a given domain, including compound angles and equations that reduce to quadratics.

Same method, new units

Example 1●

Solve $\cos x = -\dfrac{\sqrt3}{2}$ for $0 \le x \le 2\pi$.

Related angle $\dfrac{\pi}{6}$; cos is negative in quadrants 2 and 3.

$x = \pi - \dfrac{\pi}{6} = \dfrac{5\pi}{6}$ or $x = \pi + \dfrac{\pi}{6} = \dfrac{7\pi}{6}$

Solutions in radians

The x-axis runs from 0 to 2π ≈ 6.28. Each intersection is a solution.

Try this

Set $k = -0.85$ (close to $-\tfrac{\sqrt3}{2}$). Check the solutions are near $\tfrac{5\pi}{6} \approx 2.62$ and $\tfrac{7\pi}{6} \approx 3.67$.

Example 2: compound angle●●

Solve $\sin 2x = 1$ for $0 \le x \le 2\pi$.

$0 \le 2x \le 4\pi$, so $2x = \dfrac{\pi}{2}$ or $\dfrac{5\pi}{2}$.

$x = \dfrac{\pi}{4}$ or $\dfrac{5\pi}{4}$.

Example 3: a quadratic●●●

Solve $2\sin^2 x - \sin x - 1 = 0$ for $0 \le x \le 2\pi$.

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

$\sin x = 1$ gives $x = \dfrac{\pi}{2}$.

$\sin x = -\dfrac12$ gives $x = \dfrac{7\pi}{6}$ or $\dfrac{11\pi}{6}$.

Flashcards

Quadrant 2 angle

$\pi - \alpha$

Quadrant 3 angle

$\pi + \alpha$

Quadrant 4 angle

$2\pi - \alpha$

$sin 2x = k$ for $0 le x le 2pi$

Solve for $2x$ from $0$ to $4\pi$, then halve.

Practice

Solve $\sin x = \dfrac12$ for $0 \le x \le 2\pi$.
Solve $\cos x = -\dfrac{\sqrt3}{2}$ for $0 \le x \le 2\pi$.
Solve $\tan x = -1$ for $0 \le x \le 2\pi$.
Solve $2\sin^2 x - \sin x - 1 = 0$ for $0 \le x \le 2\pi$. Enter all three solutions.
Solve $\sin 2x = 1$ for $0 \le x \le 2\pi$.
To solve $\cos x = 0.3$ on a calculator, you need:
Solve $\cos x = 0.3$ for $0 \le x \le 2\pi$, to 2 decimal places.
Solve $\tan x = \sqrt3$ for $0 \le x \le 2\pi$.
Solve $\sin x = -1$ for $0 \le x \le 2\pi$.