Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• State and prove the compound angle identities for sin, cos, and tan.
• Apply these identities to simplify expressions and solve equations.
• Determine exact values for angles like 15° or 75°.
• Use the identities to prove other trigonometric identities.

Lesson summary

Trigonometry is a fundamental part of mathematics, with applications stretching far beyond the classroom. This week, we delve into compound angle identities, powerful tools that allow us to express trigonometric functions of sums or differences of angles in terms of trigonometric functions of the individual angles. Understanding these identities is crucial not only for solving complex trigonometric equations but also for simplifying expressions in various fields like physics, engineering, and even computer graphics. Imagine calculating the optimal angle for a solar panel installation considering the sun's position at different times of the year – compound angle identities come into play.

Lesson notes

The compound angle identities are a set of trigonometric equations that relate trigonometric functions of the sum or difference of two angles to trigonometric functions of the individual angles. Let's explore these identities one by one. 2.1 Sine of a Sum and Difference: Sine of a Sum: sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

Sine of a Difference: sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

Explanation: These identities state that the sine of the sum (or difference) of two angles A and B is equal to the sine of the first angle times the cosine of the second, plus (or minus) the cosine of the first angle times the sine of the second. The plus sign corresponds to the sum identity, and the minus sign to the difference identity. 2.2 Cosine of a Sum and Difference: Cosine of a Sum: cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

Cosine of a Difference: cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Explanation: Notice that for cosine, the sign changes! The cosine of the sum of two angles is the cosine of the first angle times the cosine of the second, minus the sine of the first angle times the sine of the second. Conversely, the cosine of the difference of two angles is the cosine of the first angle times the cosine of the second, plus the sine of the first angle times the sine of the second. 2.3 Tangent of a Sum and Difference: Tangent of a Sum: tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))

Tangent of a Difference: tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Explanation: The tangent identities are slightly more complex. The tangent of the sum of two angles is the sum of the tangents of the individual angles divided by one minus the product of their tangents. Similarly, the tangent of the difference of two angles is the difference of the tangents of the individual angles divided by one plus the product of their tangents. Important

Note: These identities are valid for all angles A and B where the functions are defined. Be mindful of undefined cases, especially with the tangent function.

Example 1: Finding the Exact Value of sin(75°) 75° can be expressed as 45° + 30°. We know the exact values for the trigonometric functions of 45° and 30°.

Therefore, sin(75°) = sin(45° + 30°) = sin(45°)cos(30°) + cos(45°)sin(30°) = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4 Example 2: Simplifying a Trigonometric Expression Simplify: cos(x + y) + cos(x - y)

Using the compound angle identities: cos(x + y) = cos(x)cos(y) - sin(x)sin(y) cos(x - y) = cos(x)cos(y) + sin(x)sin(y)

Adding the two expressions: cos(x + y) + cos(x - y) = [cos(x)cos(y) - sin(x)sin(y)] + [cos(x)cos(y) + sin(x)sin(y)] = 2cos(x)cos(y)

Example 3: Solving a Trigonometric Equation Solve for x: sin(x + 60°) = cos(x), where 0° ≤ x ≤ 360° Using the sine of a sum identity: sin(x + 60°) = sin(x)cos(60°) + cos(x)sin(60°) sin(x)cos(60°) + cos(x)sin(60°) = cos(x) sin(x)(1/2) + cos(x)(√3/2) = cos(x) sin(x)/2 = cos(x) - cos(x)(√3/2) sin(x)/2 = cos(x)(1 - √3/2) sin(x)/cos(x) = 2(1 - √3/2) tan(x) = 2 - √3 x = arctan(2 - √3) ≈ 15° Since tangent is positive in the first and third quadrants, we also have: x ≈ 15° + 180° = 195° Therefore, the solutions are x ≈ 15° and x ≈ 195°. Guided Practice (With Solutions)

Question 1: Expand and simplify: sin(x - 30°)

Solution: sin(x - 30°) = sin(x)cos(30°) - cos(x)sin(30°) = sin(x)(√3/2) - cos(x)(1/2) = (√3/2)sin(x) - (1/2)cos(x)

Commentary: This question directly applies the sine difference identity. We substitute the known values for cos(30°) and sin(30°) and simplify.

Question 2: Simplify: cos(2x)cos(x) + sin(2x)sin(x)

Solution: Notice that this expression matches the form of cos(A - B), where A = 2x and B = x.

Therefore, cos(2x)cos(x) + sin(2x)sin(x) = cos(2x - x) = cos(x)

Commentary: This question requires recognizing the compound angle identity in reverse. By recognizing the pattern, we can quickly simplify the expression.

Question 3: Find the exact value of tan(15°).

Solution: 15° = 45° - 30° tan(15°) = tan(45° - 30°) = (tan(45°) - tan(30°)) / (1 + tan(45°)tan(30°)) = (1 - 1/√3) / (1 + 1(1/√3)) = (√3 - 1) / (√3 + 1)

Rationalizing the denominator: = [(√3 - 1) / (√3 + 1)] * [(√3 - 1) / (√3 - 1)] = (3 - 2√3 + 1) / (3 - 1) = (4 - 2√3) / 2 = 2 - √3

Commentary: This question combines the tangent difference identity with rationalizing the denominator, a common technique. Independent Practice (Questions Only)

Expand and simplify: cos(x + 45°)

Expand and simplify: sin(2x)cos(x) - cos(2x)sin(x) Find the exact value of sin(105°)

Solve for x: cos(x - 30°) = sin(x), where 0° ≤ x ≤ 360° Prove the identity: sin(A + B) + sin(A - B) = 2sin(A)cos(B)

Simplify: [sin(A+B) - sin(A-B)] / [cos(A+B) + cos(A-B)] If sin(A) = 3/5, where A is in the first quadrant, and cos(B) = -5/13, where B is in the second quadrant, find sin(A+B). Prove that tan(x+45°) = (1 + tan(x))/(1-tan(x))

Simplify: cos(x + 60°) - cos(x - 60°)

Solve: sin(x + 30°) + cos(x + 60°) = 1 for 0° ≤ x ≤ 360°