Topic 3.10 Notes – Trigonometric Equations and Inequalities

Solving Trigonometric Equations

A trig equation asks for inputs that make something with $\sin$, $\cos$, or $\tan$ true. The main thing to remember is that inverse trig gives a principal value, which is only one angle in a restricted output range.

• $\arcsin x \in [-\pi/2,\pi/2]$
• $\arccos x \in [0,\pi]$
• $\arctan x \in (-\pi/2,\pi/2)$

So if $\arcsin(1/2)=\pi/6$, that does not mean $\pi/6$ is the only solution to $\sin x=1/2$.

Range checks

Before solving, check whether a real solution is even possible.

• $\sin u=c$ or $\cos u=c$ has no real solution if $c\notin[-1,1]$
• $\tan u=c$ has a real solution for every real $c$

General solution formulas

Because trig functions repeat, solutions repeat.

$$\text{If } \alpha=\arcsin c,\quad \sin u=c \Rightarrow u=\alpha+2\pi k \text{ or } u=\pi-\alpha+2\pi k$$

$$\text{If } \alpha=\arccos c,\quad \cos u=c \Rightarrow u=\alpha+2\pi k \text{ or } u=-\alpha+2\pi k$$

$$\text{If } \alpha=\arctan c,\quad \tan u=c \Rightarrow u=\alpha+\pi k$$

Here $k\in\mathbb Z$. Sine and cosine repeat every $2\pi$. Tangent repeats every $\pi$.

Keep the unit circle in mind when you want the other angles with the same trig value.

Unit circle with common angles

Exact and approximate answers

• Use exact unit-circle values when you can. Example: $\sin x=\sqrt{3}/2$ gives exact angles.
• $\arccos(0.3)$ is already an exact answer.
• Round only at the end.

Solving on an Interval and with Transformed Angles

When the angle is something like $3x-\pi/2$, solve for the whole angle first, then solve for $x$.

For example, solve $\cos(2x)=1/2$ on $0\le x<2\pi$.

1. Let $u=2x$
2. Solve $\cos u=1/2$, so $u=\pi/3+2\pi k$ or $u=5\pi/3+2\pi k$
3. Since $0\le x<2\pi$, you have $0\le u<4\pi$
4. List all $u$-values in that interval

   $$u=\pi/3,\ 5\pi/3,\ 7\pi/3,\ 11\pi/3$$

5. Divide by 2

   $$x=\pi/6,\ 5\pi/6,\ 7\pi/6,\ 11\pi/6$$

The spacing changes because the period changes. For $\cos(2x)$, the period in $x$ is $\pi$, not $2\pi$.

When algebra comes first

Sometimes you need algebra before inverse trig.

Example:

$$2\sin^2x-\sin x-1=0$$

Factor:

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

So $\sin x=-1/2$ or $\sin x=1$.

One common trap is dividing by a trig expression like $\sin x$. If $\sin x=0$ could be a solution, dividing would erase it.

Trigonometric Inequalities

Trig inequalities usually give intervals, not single numbers.

Basic method

1. Isolate the trig expression
2. Solve the related equation for boundary angles
3. Include undefined points as boundaries when needed
4. Use the unit circle, graph, or test values to see where the inequality is true
5. Write the answer with correct interval notation

Sine and cosine inequalities

Example:

$$\cos x\le -\frac{\sqrt2}{2}$$

Boundary angles on $[0,2\pi]$ are $3\pi/4$ and $5\pi/4$. Cosine is that small on the left side of the unit circle, so

$$x\in\left[ \frac{3\pi}{4},\frac{5\pi}{4}\right]$$

Wraparound sometimes creates two pieces, like values near both $0$ and $2\pi$.

Tangent inequalities

For tangent, think branch by branch between asymptotes.

Example:

$$\tan x>1$$

On $[0,2\pi)$, equality happens at $\pi/4$ and $5\pi/4$. Tangent is undefined at $\pi/2$ and $3\pi/2$. The graph makes it clear that you only keep the parts of each branch above $y=1$. So

$$x\in\left(\frac{\pi}{4},\frac{\pi}{2}\right)\cup\left(\frac{5\pi}{4},\frac{3\pi}{2}\right)$$

Do not apply inverse trig to both sides of an inequality as if trig functions were one-to-one on all real numbers.

Graphs, Context, and Common Mistakes

Some equations are easier with technology. If $f(x)=g(x)$, solutions are the intersection points. You can also solve $f(x)-g(x)=0$ numerically. Make sure the graphing window shows the full domain asked for.

In context, the domain often limits answers. A trig model may repeat forever mathematically, but a problem about time might only allow $0\le t\le 12$ seconds. Keep units and only report values that make sense there.

Common mistakes

• stopping after one inverse-trig output
• forgetting tangent’s period is $\pi$
• adding $2\pi$ directly to $x$ in transformed angles
• missing extra solutions over multiple cycles
• including asymptotes in tangent inequalities
• mixing radians and degrees