Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• State and prove the compound angle identities.
• Apply these identities to simplify trigonometric expressions.
• Use identities to find exact values for angles like 15° or 75°.
• Solve trigonometric equations using the identities.
• Apply these concepts to real-world problems.

Lesson summary

Trigonometry is a fundamental branch of mathematics with numerous applications in fields like surveying, navigation, engineering, and physics. Compound angle identities allow us to express trigonometric functions of sums and differences of angles in terms of trigonometric functions of the individual angles. This is crucial for simplifying complex trigonometric expressions and solving trigonometric equations, ultimately allowing us to model and solve real-world problems involving angles and distances.

Lesson notes

2.1 The Compound Angle Identities: The four fundamental compound angle identities are: sin(A + B) = sinAcosB + cosAsinB sin(A - B) = sinAcosB - cosAsinB cos(A + B) = cosAcosB - sinAsinB cos(A - B) = cosAcosB + sinAsinB 2.2 Proving the Identities: While a rigorous geometric proof is beyond the scope of this lesson note, we can illustrate the logic behind these identities. Consider the identity sin(A + B) = sinAcosB + cosAsin

B. Derivation concept (non-rigorous): Imagine a right-angled triangle. We are dealing with the sine of the SUM of two angles. The sine (opposite/hypotenuse) of the combined angle (A+B) can be broken down into the contributions from angle A and angle B. When the angle is ‘A+B’, the 'opposite' side can be related to the sines and cosines of the individual angles A and B. Similarly, the other identities can be derived conceptually, but the key is memorizing the formulas to apply effectively. 2.3 Using the Identities: The power of these identities lies in their ability to rewrite trigonometric expressions. Let's look at some examples.

Example 1: Finding the exact value of sin 75° We can express 75° as 45° + 30°.

Therefore, we can use the sin(A + B) identity: sin 75° = sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4 Therefore, sin 75° = (√6 + √2) /

4. Example 2: Simplifying a trigonometric expression Simplify: cos(x + y) + cos(x - y)

Using the identities: cos(x + y) = cosx cosy - sinx siny cos(x - y) = cosx cosy + sinx siny Therefore, cos(x + y) + cos(x - y) = (cosx cosy - sinx siny) + (cosx cosy + sinx siny) = 2cosx cosy Example 3: Solving a Trigonometric Equation Solve for x: sin(x + 30°) = cos x for 0° ≤ x ≤ 360° Using the sin(A + B) identity: sin x cos 30° + cos x sin 30° = cos x (√3/2)sin x + (1/2)cos x = cos x (√3/2)sin x = (1/2)cos x √3 sin x = cos x tan x = 1/√3 x = 30° or x = 210° (since tangent is positive in the first and third quadrants).

Example 4: Using identities to prove other identities: Prove that sin(90° + x) = cos x Using the sin(A + B) identity: sin(90° + x) = sin 90° cos x + cos 90° sin x Since sin 90° = 1 and cos 90° = 0: sin(90° + x) = (1) cos x + (0) sin x sin(90° + x) = cos x Guided Practice (With Solutions)

Question 1: Expand and simplify: cos(45° - x)

Solution: cos(45° - x) = cos 45° cos x + sin 45° sin x = (√2/2) cos x + (√2/2) sin x = (√2/2)(cos x + sin x)

Commentary: This question directly applies the cos(A - B) identity. The key is to correctly substitute the values of cos 45° and sin 45°.

Question 2: Find the exact value of sin 15° Solution: sin 15° = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2) / 4

Commentary: This question uses the sin(A - B) identity. Choosing suitable angles (45° and 30°) whose trigonometric values are known is crucial.

Question 3: Simplify: sin(x + y) - sin(x - y)

Solution: sin(x + y) = sin x cos y + cos x sin y sin(x - y) = sin x cos y - cos x sin y sin(x + y) - sin(x - y) = (sin x cos y + cos x sin y) - (sin x cos y - cos x sin y) = 2 cos x sin y

Commentary: This question involves simplifying by expanding and then canceling terms. Understanding the signs in the identities is important.

Question 4: Determine the general solution of the equation: cos(x + 60°) = sin x Solution: cos(x + 60°) = cos x cos 60° - sin x sin 60° = sin x (1/2) cos x - (√3/2) sin x = sin x (1/2) cos x = (1 + √3/2) sin x cos x = (2 + √3)sin x (1/(2+√3)) = tan x Rationalizing gives tan x = 2 - √3 Therefore, x = arctan(2-√3) + k.180, k ∈ Z x ≈ 15° + k.180, k ∈ Z

Commentary: This question combines the compound angle identities with solving trigonometric equations. Rationalizing denominators can be helpful. Independent Practice (Questions Only)

Expand and simplify: sin(x - 60°) Find the exact value of cos 105° Simplify: cos(a + b) - cos(a - b)

Prove the identity: cos(90° - x) = sin x Solve for x: sin(x - 45°) = cos x, for 0° ≤ x ≤ 360° Express sin(3x) in terms of sin(x). (Hint: sin(3x) = sin(2x + x))

Prove: `(sin(A+B))/(cos(A)cos(B)) = tan(A) + tan(B)` If sin A = 3/5, where A is an acute angle, and cos B = 5/13, where B is an acute angle, find the exact value of sin(A + B).

Simplify the expression: `cos(x+45°)cos(x-45°) - sin(x+45°)sin(x-45°)` Show that tan(A + B) = (tan A + tan B) / (1 - tan A tan B) [HINT: use sin(A+B) / cos(A+B)]