General Solutions of Trigonometric Equations (Radians)

Introduction

This post explains how to find the general solutions of trigonometric equations in radians. The method is identical to the degrees version: find the principal value and then use the appropriate general solution formula for sine, cosine or tangent. These examples are ideal for A-level Maths and Further Maths students who need to work confidently in radians.

Remember:

π radians = 180°

In radian mode your calculator can give solutions either as decimals or in terms of π. For general solutions, we use the following standard formulas:

Where PV is the principal value returned by the inverse trig function, and n ∈ ℤ.

Examples Using Radians

Example 1 — Sine Equation

Solve for 0≦x≦π the equation

Using the sine formula,

n=0 ⇒ x=1.74 rad

n=1 ⇒ x=2.97 rad

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Example 2 — Cosine Equation

Giving your answers as multiples of π, solve for 0≦x≦2π the equation

Using the cosine formula,

n=1 ⇒ x=23/24π, 19/24π

n=2 ⇒ x=47/24π, 43/24π

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Example 3 — Tangent Equation

Giving your answers as multiples of π, solve for 0≦x≦4π the equation

Using the tan formula,

n=0 ⇒ x=π/3

n=1 ⇒ x=7π/3, 5π/3

n=2 ⇒ x=11π/3

Conclusion

General solutions in radians follow the same pattern as the degree-based method: calculate the principal value, then apply the standard formulas for sine, cosine or tangent to capture every solution in the required interval.

If you’d like to compare with examples in degrees, see my companion post: General Solutions of Trigonometric Equations (Degrees).