Trigonometric equations ask you to find angles that make a sine, cosine, tangent, or related expression true. They matter because periodic motion, waves, circular motion, and rotations often repeat in predictable patterns. Solving these equations requires both algebra skills and a strong understanding of the unit circle.
The main goal is to find all angles that work, not just the first one shown by a calculator.
Key Facts
- sin x and cos x repeat every 2π radians: sin(x + 2πk) = sin x and cos(x + 2πk) = cos x
- tan x repeats every π radians: tan(x + πk) = tan x
- If sin x = a, then solutions in one cycle come from the unit circle, then repeat as x + 2πk
- If cos x = a, then solutions in one cycle come from the x-coordinates on the unit circle, then repeat as x + 2πk
- Use identities to rewrite equations, such as sin^2 x + cos^2 x = 1 and tan x = sin x / cos x
- Always check the requested interval, such as 0 ≤ x < 2π, because it controls which repeated solutions are included
Vocabulary
- Trigonometric equation
- An equation that contains one or more trigonometric functions of a variable angle.
- Unit circle
- A circle of radius 1 centered at the origin that connects angles to cosine and sine values.
- Period
- The horizontal length after which a trigonometric function repeats its values.
- General solution
- A formula that lists every solution of a trigonometric equation using an integer variable such as k.
- Reference angle
- The acute angle between the terminal side of an angle and the x-axis.
Common Mistakes to Avoid
- Stopping after one calculator answer is wrong because trigonometric functions repeat and usually have infinitely many solutions.
- Forgetting quadrant signs is wrong because sine, cosine, and tangent are positive or negative in different quadrants.
- Dividing by a trig expression without checking whether it can be zero is wrong because it may remove valid solutions from the equation.
- Mixing degrees and radians is wrong because values like 30 and π/6 represent different angle measures unless the mode is handled correctly.
Practice Questions
- 1 Solve sin x = 1/2 for 0 ≤ x < 2π.
- 2 Solve 2cos^2 x - 1 = 0 for 0 ≤ x < 2π.
- 3 Explain why the equation tan x = 1 has solutions that repeat every π instead of every 2π.