Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Solve trigonometric equations for a specified interval.
• Verify and prove trigonometric identities.
• Apply identities to simplify complex expressions.
• Identify the correct identities for specific problems.
• Relate equation solutions to trigonometric graphs.

Lesson summary

This week, we delve deeper into the fascinating world of Trigonometry, building upon the foundational concepts learned earlier in the term. We will be focusing specifically on solving trigonometric equations and proving trigonometric identities. Trigonometry is essential not just for passing Mathematics, but also for various practical fields such as surveying, navigation, engineering (especially civil engineering related to infrastructure development in South Africa), and even aspects of computer graphics and animation. Understanding trigonometric relationships allows us to solve real-world problems involving angles and distances, critical for accurate measurements and constructions.

Lesson notes

Solving Trigonometric Equations A trigonometric equation is an equation that contains trigonometric functions. The goal is to find the values of the angle (usually represented by x or θ) that satisfy the equation. Unlike trigonometric identities, which are true for all values of the variable, trigonometric equations are only true for specific values or a range of values.

General Approach: Isolate the Trigonometric Function: Use algebraic manipulation to isolate the trigonometric function (sin x, cos x, tan x, etc.) on one side of the equation.

Find the Reference Angle: Determine the reference angle, which is the acute angle whose trigonometric value (without the sign) is equal to the value obtained in step

1. Use special triangles (30-60-90, 45-45-90) or a calculator to find the reference angle.

Determine the Quadrants: Based on the sign of the trigonometric function, determine the quadrants where the solutions lie.

Remember the CAST rule (Quadrant I: All positive, Quadrant II: Sine positive, Quadrant III: Tangent positive, Quadrant IV: Cosine positive).

Find the General Solution: Write down the general solution for the angle x. The general solution gives all possible solutions to the equation, considering the periodic nature of trigonometric functions.

The general solutions are as follows: sin x = sin α => x = α + k.360° or x = (180° - α) + k.360°, where k ∈ Z (set of integers). cos x = cos α => x = α + k.360° or x = -α + k.360°, where k ∈ Z. tan x = tan α => x = α + k.180°, where k ∈

Z. Find Specific Solutions: If a specific interval is given (e.g., 0° ≤ x ≤ 360°), find all the solutions that lie within that interval by substituting different integer values for k in the general solution.