← Precalculus
How to solve
Solving sin θ = k, cos θ = k, or tan θ = k means finding every angle on the circle whose y-coordinate, x-coordinate, or ratio equals k. Start from a reference solution, then include every matching angle in [0, 2π). Most of these equations have two solutions per rotation, but sin θ = 1, sin θ = −1, cos θ = 1, and cos θ = −1 (the curve's extremes) have exactly one.
Steps
- Find one reference solution from the unit circle.
- Use symmetry to list all solutions in [0, 2π) with the same sine, cosine, or tangent.
- A few boundary values have only one solution in the interval — enter it alone.
- Order multi-solution answers as the drill requires (often smaller first).
Example
Solve sin θ = 1/2 on [0, 2π).
- Reference angle π/6 has sine 1/2.
- In [0, 2π), sine is also 1/2 at π − π/6 = 5π/6.
- Solutions: π/6, 5π/6.
Answer: pi/6, 5pi/6
How to type answers
- Enter angles in terms of pi, except zero — write that solution as plain 0, not 0pi. If there are two, put the smaller one first, like pi/6, 5pi/6; if there is only one, enter it alone, like pi/2.
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