medium difficulty8 min

Simplifying and Solving Equations with Inverse Trigonometric Functions

Simplification and equation solving with inverse trigonometric functions means reducing expressions or finding variable values while preserving domain restrictions and principal value conditions.

In one line: Solve first, but trust only the values that survive the domain and original-equation check.

Practice This Concept

Understand it

Inverse trigonometric equations often look algebraic after applying a trigonometric function to both sides, but this can introduce extra answers. Every candidate must be checked in the original equation and in the domain of each inverse function.

Example

If sin⁻¹x = π/6, then x = sin(π/6) = 1/2. Since 1/2 ∈ [−1,1], it is valid.

Formula and rule

Method rules: preserve domains such as x ∈ [−1,1] for sin⁻¹x and cos⁻¹x; use identities only under their conditions; when applying trig functions to both sides, check principal ranges; for equations involving tan⁻¹, verify denominator conditions in sum and difference formulae.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Solve 2sin⁻¹x = cos⁻¹x.

1Since sin⁻¹x + cos⁻¹x = π/2, write cos⁻¹x = π/2 − sin⁻¹x.
2Then 2sin⁻¹x = π/2 − sin⁻¹x, so 3sin⁻¹x = π/2.
3Hence sin⁻¹x = π/6 and x = sin(π/6) = 1/2.
4Check: 2sin⁻¹(1/2) = 2 × π/6 = π/3 and cos⁻¹(1/2) = π/3.
5Therefore x = 1/2.

Method you can reuse

List domain restrictions, simplify using valid identities, solve the resulting algebraic or trigonometric statement, substitute candidate values into the original equation, and reject any value outside the allowed domain or principal branch.

Visual tip

Usually not required unless the equation asks for graphical interpretation. A number line for domain restrictions is useful for filtering candidates.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Solve sin⁻¹x = cos⁻¹x.

Think first, then show answer

Using sin⁻¹x + cos⁻¹x = π/2, let both be equal. Then 2sin⁻¹x = π/2, so sin⁻¹x = π/4 and x = √2/2.

Common wrong answer

Forgetting to check the original equation after squaring or applying a trigonometric function, leading to an extra value of x.

Don't mix this up

Taking sine, cosine or tangent on both sides and accepting every algebraic solution without checking the original inverse trigonometric equation.

Exam tip

Write the allowed interval for x before solving. The final verification step is often the difference between a correct answer and an extraneous root.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions appear as simplify, prove, evaluate, or solve for x, often combining identities, branch restrictions and exact angle values.

1
Enough for 1 mark
Simplification and equation solving with inverse trigonometric functions means reducing expressions or finding variable values while preserving domain restrictions and principal value conditions.
2
Add for 2 marks
Example: If sin⁻¹x = π/6, then x = sin(π/6) = 1/2. Since 1/2 ∈ [−1,1], it is valid.
3
Add for 3 marks
Inverse trigonometric equations often look algebraic after applying a trigonometric function to both sides, but this can introduce extra answers.

MCQ Quiz

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