Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• State and apply the compound angle identities for sin, cos, and tan.
• Simplify trigonometric expressions.
• Solve trigonometric equations.
• Prove trigonometric identities.
• Apply these identities in problem-solving.

Lesson summary

Compound angle identities form a crucial part of trigonometry, allowing us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. This is a powerful tool that simplifies complex trigonometric expressions and solves equations that would otherwise be intractable. In South Africa, these identities are essential for various applications, from surveying and navigation to engineering and the design of structures that must withstand specific forces at angles. Understanding compound angle identities provides a fundamental building block for further studies in calculus and physics.

Lesson notes

The compound angle identities are as follows: Sine of a Sum/Difference: sin(A + B) = sinA cosB + cosA sinB sin(A - B) = sinA cosB - cosA sinB Cosine of a Sum/Difference: cos(A + B) = cosA cosB - sinA sinB cos(A - B) = cosA cosB + sinA sinB Tangent of a Sum/Difference: tan(A + B) = (tanA + tanB) / (1 - tanA tanB) tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

Explanation: These identities arise from geometric considerations on the unit circle and are derived through careful manipulation of angles and their associated trigonometric ratios. It's crucial to understand why these identities hold true, not just memorize them. Consider a unit circle. Drawing angles A and B, and A+B, and constructing right triangles will allow one to geometrically derive the relationships.