Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Solve trigonometric equations for a specific domain.
• Find the general solutions of trigonometric equations.
• Verify and apply trigonometric identities.
• Use compound and double angle formulae to solve equations.

Lesson summary

Trigonometry is a fundamental branch of mathematics with wide-ranging applications, from surveying land for housing projects in our communities to designing stable bridges and buildings. Understanding trigonometric equations and identities is crucial for careers in engineering, architecture, navigation, and even game development. This week, we will delve into solving trigonometric equations, verifying trigonometric identities, and applying these skills to solve problems. Mastering these concepts will not only improve your performance in Mathematics but also equip you with valuable problem-solving skills applicable to diverse fields.

Lesson notes

2.1 Solving Trigonometric Equations Solving trigonometric equations involves finding the values of the angle (usually represented by x, θ, or some other variable) that satisfy the given equation. This process uses algebraic manipulation, inverse trigonometric functions, and understanding the periodicity of trigonometric functions.

Steps to Solve Trigonometric Equations: Isolate the trigonometric function: Manipulate the equation algebraically to isolate the trigonometric function (e.g., sin x, cos θ, tan α) on one side.

Find the reference angle: Determine the reference angle (the acute angle formed between the terminal arm of the angle and the x-axis) using the inverse trigonometric function (e.g., arcsin, arccos, arctan). The reference angle is always positive and acute (between 0° and 90°).

Determine the quadrants: Identify the quadrants in which the trigonometric function has the correct sign (positive or negative) according to the original equation. Remember the CAST diagram (C-Cos positive in the 4th quadrant, A-All positive in the 1st quadrant, S-Sin positive in the 2nd quadrant, T-Tan positive in the 3rd quadrant).

Find the angles in the specified interval: Use the reference angle and the quadrants identified in step 3 to find all angles within the given interval that satisfy the equation.

General Solutions: If the interval is not specified, find the general solution by adding integer multiples of the period of the trigonometric function to the solutions found in step

4. The period of sin x and cos x is 360°, while the period of tan x is 180°. 2.2 Trigonometric Identities Trigonometric identities are equations that are true for all values of the variable(s) for which the expressions are defined. They are used to simplify trigonometric expressions and solve trigonometric equations.

Fundamental Identities: Reciprocal Identities: cosec θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ Quotient Identities: tan θ = sin θ/cos θ cot θ = cos θ/sin θ Pythagorean Identity: sin² θ + cos² θ = 1 1 + tan² θ = sec² θ 1 + cot² θ = cosec² θ Compound Angle Formulae: sin (A + B) = sin A cos B + cos A sin B sin (A - B) = sin A cos B - cos A sin B cos (A + B) = cos A cos B - sin A sin B cos (A - B) = cos A cos B + sin A sin B tan (A + B) = (tan A + tan B) / (1 - tan A tan B) tan (A - B) = (tan A - tan B) / (1 + tan A tan B)

Double Angle Formulae: sin 2A = 2 sin A cos A cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A tan 2A = (2 tan A) / (1 - tan² A) 2.3 Verifying Trigonometric Identities To verify a trigonometric identity, you must show that one side of the equation can be transformed into the other side using known trigonometric identities and algebraic manipulations.

Steps to Verify Trigonometric Identities: Choose a side: Select the more complicated side of the equation to work with. It's usually easier to simplify a complex expression than to make a simple expression more complex. Apply identities and algebraic manipulations: Use known trigonometric identities, algebraic techniques (e.g., factoring, expanding, simplifying fractions), and creative problem-solving to manipulate the chosen side.

Stop when the sides are equal: Continue manipulating until the chosen side is identical to the other side of the equation.

Worked example

Example 1: Solving a Trigonometric Equation

Solve the equation 2sin x - 1 = 0 for 0° ≤ x ≤ 360°.

Solution:

Isolate the trigonometric function:

2sin x = 1

sin x = 1/2

Find the reference angle:

x ref = arcsin(1/2) = 30°

Determine the quadrants:

Since sin x is positive, x is in the 1st and 2nd quadrants.

Find the angles in the specified interval:

1st quadrant: x = 30°

2nd quadrant: x = 180° - 30° = 150°

Solution set: {30°, 150°}

Example 2: Finding the General Solution of a Trigonometric Equation