Trig Substitution in Algebra

Key idea

Use trig to solve algebraic inequalities.

In depth

Trig substitution can prove inequalities or solve problems with constraints like x² + y² = 1 by writing x = cos θ, y = sin θ. Then an expression such as x + y becomes cos θ + sin θ ≤ √2.

Worked example

Maximise x + y subject to x² + y² = 1: cos θ + sin θ = √2 sin(θ + 45°) has maximum √2.

Real-world uses

• Solving inequalities and integrals in advanced engineering analysis

Common mistake

Using a substitution that does not cover all of the allowed values.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Trigonometry

• Half Angle Formulas – Trig of θ/2.
• Power Reducing Formulas – Reduce squares to first powers of cos 2θ.
• Product-to-Sum Formulas – Convert products into sums.
• Sum-to-Product Formulas – Convert sums into products.
• Triple Angle Formulas – sin3θ and cos3θ in terms of θ.
• Trig and Inverse Trig Compositions – Evaluate expressions like sin(arccos x).