Solving Simple Trigonometric Inequalities
What Are Trigonometric Inequalities?
A trigonometric inequality is an inequality that involves trigonometric functions such as $\sin x$, $\cos x$, or $\tan x$. Solving them means finding all values of $x$ that make the inequality true.
Examples of simple trigonometric inequalities:
- $\sin x > \frac{1}{2}$
- $\cos x \leq -\frac{\sqrt{2}}{2}$
- $\tan x < 1$
The Key Strategy: Using the Unit Circle
The fundamental approach to solving trigonometric inequalities involves three steps:
Step 1
Solve the equality
Replace the inequality sign with $=$ and find the boundary angles where the trig function equals the given value.
Step 2
Mark on the unit circle
Identify the boundary points on the unit circle and determine which arcs satisfy the inequality.
Step 3
Write the solution
Express the solution as intervals, adding $+ 2\pi k$ (where $k \in \mathbb{Z}$) for the general solution.