Basic trigonometric equations

The solutions of $\displaystyle {\sin x}={a}$ are the numbers whose point on the unit circle is at height $\displaystyle a$. On the board, $\displaystyle {a}={\frac{\sqrt{3}}{2}}$: the line $\displaystyle {y}={\frac{\sqrt{3}}{2}}$ cuts the circle at two points.

• $\displaystyle {A}={P\left(\frac{\pi}{3}\right)}$, and $\displaystyle B$ is its reflection in the $\displaystyle y$-axis: $\displaystyle {B}={P\left(\pi-\frac{\pi}{3}\right)}={P\left(\frac{2\pi}{3}\right)}$.
• Each point belongs to infinitely many numbers, $\displaystyle +2\pi k$.

So, when $\displaystyle {\sin\alpha}={a}$:

This is the general solution: all the solutions, $\displaystyle k$ an integer. In the example: $\displaystyle {x}={\frac{\pi}{3}+2\pi k}$ or $\displaystyle {x}={\frac{2\pi}{3}+2\pi k}$.

• When $\displaystyle |a|>1$ the line does not meet the circle: there is no solution.
• When $\displaystyle {a}={1}$, $\displaystyle {a}={-1}$ or $\displaystyle {a}={0}$ the two families give the same points: $\displaystyle {\sin x}={1}$ – $\displaystyle {x}={\frac{\pi}{2}+2\pi k}$; $\displaystyle {\sin x}={0}$ – $\displaystyle {x}={\pi k}$.

The solutions of $\displaystyle {\cos x}={b}$ are the numbers whose point is on the upright line $\displaystyle {x}={b}$. The two points are reflections of each other in the $\displaystyle x$-axis: $\displaystyle P(\alpha)$ and $\displaystyle P(-\alpha)$. When $\displaystyle {\cos\alpha}={b}$:

For example, $\displaystyle {\cos x}={-\frac{1}{2}}$: $\displaystyle {\cos\frac{2\pi}{3}}={-\frac{1}{2}}$, so $\displaystyle {x}={\pm\frac{2\pi}{3}+2\pi k}$. When $\displaystyle |b|>1$ there is no solution.

Every number $\displaystyle c$ has a point $\displaystyle (1,c)$ on the tangent line, and the line from the origin through it cuts the circle at two opposite points – half a turn apart. When $\displaystyle {\tan\alpha}={c}$:

For example, $\displaystyle {\tan x}={-1}$: $\displaystyle {x}={-\frac{\pi}{4}+\pi k}$. This equation has a solution for every $\displaystyle c$.

When $\displaystyle a$ is not one of the special values, the calculator (set to radians) gives $\displaystyle \alpha$:

• $\displaystyle \sin^{-1}a$ – the number between $\displaystyle -\frac{\pi}{2}$ and $\displaystyle \frac{\pi}{2}$ whose sine is $\displaystyle a$;
• $\displaystyle \cos^{-1}b$ – the number between $\displaystyle 0$ and $\displaystyle \pi$ whose cosine is $\displaystyle b$;
• $\displaystyle \tan^{-1}c$ – the number between $\displaystyle -\frac{\pi}{2}$ and $\displaystyle \frac{\pi}{2}$ whose tangent is $\displaystyle c$.

The calculator gives one solution only; the formulas give the rest. For example, $\displaystyle {\sin x}={0.3}$:

• $\displaystyle {x}={\sin^{-1}0.3+2\pi k}$ or $\displaystyle {x}={\pi-\sin^{-1}0.3+2\pi k}$;
• approximately: $\displaystyle x\approx0.305+2\pi k$ or $\displaystyle x\approx2.837+2\pi k$.

To find the solutions in a given interval, write the general solution and substitute values of $\displaystyle k$ until you leave the interval.

The example below solves $\displaystyle {\cos x}={-\frac{1}{2}}$ in the interval $\displaystyle 0\leq x\leq2\pi$.

• The general solution: $\displaystyle {x}={\frac{2\pi}{3}+2\pi k}$ or $\displaystyle {x}={-\frac{2\pi}{3}+2\pi k}$.
• In the first family, $\displaystyle {k}={0}$ gives $\displaystyle \frac{2\pi}{3}$; $\displaystyle {k}={1}$ is already greater than $\displaystyle 2\pi$.
• In the second family, $\displaystyle {k}={0}$ gives a negative number; $\displaystyle {k}={1}$ gives $\displaystyle \frac{4\pi}{3}$.

The solutions: $\displaystyle {x}={\frac{2\pi}{3}}$ or $\displaystyle {x}={\frac{4\pi}{3}}$.

Solve. Write the general solution.

Solve with $\displaystyle \sin^{-1}$ and $\displaystyle \tan^{-1}$. Write the general solution.

Solve in the given interval.

$\displaystyle 0\leq x\leq2\pi$

$\displaystyle -\pi\leq x\leq\pi$

$\displaystyle 0\leq x\leq4\pi$