Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Simplify trigonometric expressions using identities.
• Prove more complex trigonometric identities.
• Solve trigonometric equations within a specific interval.
• Apply this knowledge to solve contextual problems.

Lesson summary

This week, we delve deeper into trigonometric identities and equations. Building upon our knowledge of basic trigonometric ratios (sine, cosine, tangent, cosecant, secant, and cotangent) and their relationships, we will explore more complex identities and learn how to solve trigonometric equations. This is crucial because trigonometric principles are fundamental in various fields, from surveying and navigation to engineering and physics. In the South African context, understanding trigonometry is vital in fields like civil engineering for building roads and bridges, in surveying for land management, and in telecommunications for signal transmission. Specifically, we will focus on:

Lesson notes

Fundamental Trigonometric Identities: These are the building blocks of trigonometric manipulation. It's essential to memorize and understand them: Reciprocal Identities: csc 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 Identities: sin² θ + cos² θ = 1 1 + tan² θ = sec² θ 1 + cot² θ = csc² θ Why these identities are important: They allow us to rewrite trigonometric expressions in different forms, which is crucial for simplifying expressions and solving equations. Think of them as different languages to express the same trigonometric truth.

Angle Sum and Difference Identities: 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)

Why these identities are important: They enable us to find the trigonometric values of angles that are not standard angles (0°, 30°, 45°, 60°, 90°, etc.) by expressing them as sums or differences of known angles. For example, we can find sin(75°) by using sin(45° + 30°).

Double Angle Identities: sin 2θ = 2 sin θ cos θ cos 2θ = cos² θ - sin² θ = 2cos² θ - 1 = 1 - 2sin² θ tan 2θ = (2 tan θ) / (1 - tan² θ)

Why these identities are important: They provide a direct way to relate trigonometric values of an angle to the trigonometric values of twice that angle. This is particularly useful in solving equations and simplifying expressions.

Solving Trigonometric Equations: Solving trigonometric equations involves finding the values of the variable (usually θ or x) that satisfy the equation.

Key steps include: Isolate the trigonometric function: Use algebraic manipulation to get the trigonometric function (e.g., sin θ, cos θ, tan θ) by itself on one side of the equation.

Find the reference angle: Determine the angle (usually between 0° and 90°) whose trigonometric value is equal to the absolute value of the trigonometric function on the other side of the equation. Use CAST diagram!

Determine the quadrants: Identify the quadrants in which the trigonometric function has the correct sign (positive or negative). This is where the CAST diagram comes in handy.

Find the general solution: Write the general solution for each quadrant, considering the periodicity of the trigonometric function (360° for sin and cos, 180° for tan). Find the specific solutions within the given interval: If the question specifies an interval (e.g., 0° ≤ θ ≤ 360°), find all the solutions that fall within that interval.

Worked example

Example 1: Simplifying a Trigonometric Expression

Simplify: (sin θ / cos θ) + (cos θ / sin θ)

Solution:

Find a common denominator: (sin² θ + cos² θ) / (sin θ cos θ)

Apply the Pythagorean identity (sin² θ + cos² θ = 1): 1 / (sin θ cos θ)

Rewrite using reciprocal identities: (1/sin θ) * (1/cos θ) = csc θ sec θ

Therefore, (sin θ / cos θ) + (cos θ / sin θ) = csc θ sec θ