Solving Trigonometric Equations
Basic Trigonometric Equations
Trigonometric equations involve trigonometric functions and are solved by finding the angles that satisfy the given condition.
Linear Trigonometric Equations
The simplest form of trigonometric equations are linear, such as:
$2 \sin x = 1, 0 \leq x \leq 2\pi$
To solve this:
- Isolate the trigonometric function: $\sin x = \frac{1}{2}$
- Find the reference angle: $x = \arcsin(\frac{1}{2}) = \frac{\pi}{6}$
- Identify all solutions in the given interval:
- $x = \frac{\pi}{6}$ (in the first quadrant)
- $x = \pi - \frac{\pi}{6} = \frac{5\pi}{6}$ (in the second quadrant)
For the equation $2 \sin 2x = 3 \cos x, 0° \leq x \leq 180°$:
- Rearrange: $2 \sin 2x - 3 \cos x = 0$
- Use the double angle formula: $2(2 \sin x \cos x) - 3 \cos x = 0$
- Factor out $\cos x$: $\cos x(4 \sin x - 3) = 0$
- Solve each factor:
$\cos x = 0 \to x = 90°$
$4\sin x - 3 = 0\to x = \arcsin\frac34 \to x\approx48.6°$
Therefore, the solutions are approximately 48.6° and 90°.
Common MistakeDon't forget to pay attention to the possible domain of $x$. For example, consider the following equation:
$\sin2x = \frac12, 0 \leq x \leq 2\pi$
If $0 \leq x \leq 2\pi$, that means $0 \leq 2x \leq 4\pi$! So the possible values of $2x$ are $\frac\pi6$, $\frac{5\pi}6$, $\frac{13\pi}6$, and $\frac{17\pi}6$, leading to:
$x = \frac{\pi}{12}$ or $x = \frac{5\pi}{12}$ or $x = \frac{13\pi}{12}$ or $x = \frac{17\pi}{12}$
If you see that $0 \leq x \leq \pi$ and assume that also applies to $2x$, you'll only get two of the solutions, and lose the final answer mark.
Equations with Multiple Angles
Equations involving multiple angles or combinations of trigonometric functions require careful manipulation before solving.
ExampleFor $2 \tan(3(x-4)) = 1, -\pi \leq x \leq 3\pi$:
- Simplify: $\tan(3(x-4)) = \frac{1}{2}$
- Solve for the inner expression: $3(x-4) = \arctan(\frac{1}{2}) + n\pi$, where $n$ is an integer
- Solve for $x$: $x = \frac{\arctan(\frac{1}{2})}{3} + \frac{n\pi}{3} + 4$
- Find values of $n$ that give solutions in the interval $[-\pi, 3\pi]$
This yields multiple solutions within the given interval.
Quadratic Trigonometric Equations
Some trigonometric equations lead to quadratic equations in $\sin x$, $\cos x$, or $\tan x$. These are solved using standard quadratic equation techniques.
Quadratic in Sine or Cosine
For equations like $2 \sin^2 x + 5 \cos x + 1 = 0$ for $0 \leq x \leq 4\pi$:
- Substitute $u = \cos x$: $2(1-u^2) + 5u + 1 = 0$
