Mastering Trigonometric Identities and Equations

• newEditor

Mastering Trigonometric Identities and Equations

Learning Objectives

At the end of the lesson, you will be able to calculate trigonometric ratios, know exact values for 30°, 45°, 60°, use identities, solve simple and complex trigonometric equations, and handle quadratic trigonometric equations.

What do you already know about trigonometric identities and equations?

Introduction to Trigonometric Ratios

Sine, cosine, and tangent are fundamental trigonometric ratios. They relate the angles and sides of right-angled triangles.

Exact Trigonometric Ratios

Memorise: sin(30°)=1/2, cos(30°)=√3/2, tan(30°)=1/√3; sin(45°)=√2/2, cos(45°)=√2/2, tan(45°)=1; sin(60°)=√3/2, cos(60°)=1/2, tan(60°)=√3.

The Relationship tanθ = sinθ/cosθ

The tangent of an angle is the ratio of its sine to its cosine. tanθ = sinθ/cosθ.

The Pythagorean Identity

The identity sin²θ + cos²θ = 1 holds for any angle θ. This is a fundamental trigonometric identity.

Solving sinθ = k

To solve sinθ = k, find the primary angle θ₁ and use periodicity: θ = θ₁ + 360°n or θ = 180° - θ₁ + 360°n.

Solving cosθ = k

To solve cosθ = k, find the primary angle θ₁: θ = ±θ₁ + 360°n.

Solving tanθ = k

To solve tanθ = k, find the principal value θ₁: θ = θ₁ + 180°n.

Complex Equations: sinnθ = k

Solve by finding angles using periodicity: θ = θ₁/n + 360°n or θ = (180° - θ₁)/n + 360°n.

Complex Equations: sin(θ±α) = k

Use angle addition/subtraction formulas to simplify and solve.

Solving cos(θ±α) = k

Apply angle addition/subtraction formulas to find solutions.

Solving tan(θ±α) = k

Use angle addition/subtraction formulas to simplify tan expressions.

Quadratic Trigonometric Equations

Equations like a*sin²θ + b*sinθ + c = 0 can be solved using quadratic techniques.

Review and Practice

Revisit key concepts and identities. Practice with exercises involving each type of equation discussed.

Write down 3 things you learned in this lesson.

Write down 2 things you want to know more about.

Ask 1 question about something you haven't quite understood yet.