How to Use the Double and Half Angle Formulas for Trigonometry (Precalculus - Trigonometry 28)

Overview

Professor Leonard explains the application of double and half-angle formulas in trigonometry, demonstrating how to derive sine, cosine, and tangent values for angles like 2θ and θ/2 given initial conditions. He emphasizes the critical role of quadrant identification in determining the correct sign for half-angle formulas and illustrates how these identities can be used to find exact trigonometric values for non-unit circle angles and solve trigonometric equations.

Key takeaways

Quadrant analysis is crucial for determining the correct sign (positive or negative) when using half-angle formulas.
Double-angle formulas can be used to simplify trigonometric expressions and solve equations by transforming angles (e.g., 2θ to θ).
Trigonometric identities are bidirectional; they can be used to expand or condense expressions, aiding in proofs and problem-solving.
When solving trigonometric equations involving double angles, choose the formula that best matches the other terms in the equation to facilitate factoring or simplification.
Half-angle formulas allow for the calculation of exact trigonometric values for angles not typically found on the unit circle (e.g., 22.5°, π/8) by relating them to angles that are on the unit circle.

Chapters

0:00 Introduction to Double and Half-Angle Formulas

Focus on applying previously proven double and half-angle formulas for sine, cosine, and tangent.
Key challenges include finding unknown sine/cosine values and determining the quadrant for θ/2.
Given one trig function and its quadrant, all others can be found.

2:01 Finding Sine and Tangent from Cosine in Quadrant 1

Example: Given cos(θ) = 3/5 and θ in Quadrant 1 (0 to π/2).
Using Pythagorean theorem (x² + y² = r²) with x=3, r=5 to find y=4.
Since θ is in Quadrant 1, y is positive, so sin(θ) = 4/5 and tan(θ) = 4/3.

8:22 Calculating Sine of Double Angle (2θ)

Formula: sin(2θ) = 2 sin(θ) cos(θ).
Substitute known values: sin(θ) = 4/5, cos(θ) = 3/5.
Calculation: 2 * (4/5) * (3/5) = 24/25.

13:21 Calculating Cosine of Double Angle (2θ)

Three formulas available: cos²θ - sin²θ, 1 - 2sin²θ, 2cos²θ - 1.
Using 2cos²θ - 1: 2 * (3/5)² - 1 = 2 * (9/25) - 1 = 18/25 - 25/25 = -7/25.
Using cos²θ - sin²θ: (3/5)² - (4/5)² = 9/25 - 16/25 = -7/25.

20:09 Calculating Sine of Half Angle (θ/2)

Formula: sin(θ/2) = ±√((1 - cos(θ))/2).
Substitute cos(θ) = 3/5: ±√((1 - 3/5)/2) = ±√((2/5)/2) = ±√(1/5) = ±√5/5.
Determine sign: θ in Q1 (0 to π/2) means θ/2 in Q1 (0 to π/4), so sin(θ/2) is positive.

28:49 Calculating Cosine of Half Angle (θ/2)

Formula: cos(θ/2) = ±√((1 + cos(θ))/2).
Substitute cos(θ) = 3/5: ±√((1 + 3/5)/2) = ±√((8/5)/2) = ±√(4/5) = ±2/√5 = ±2√5/5.
Determine sign: θ/2 is in Q1, so cos(θ/2) is positive.

33:27 Calculating Tangent of Double Angle (2θ)

Formula: tan(2θ) = 2tan(θ) / (1 - tan²(θ)).
Substitute tan(θ) = 4/3: (2 * 4/3) / (1 - (4/3)²) = (8/3) / (1 - 16/9) = (8/3) / (-7/9).
Simplify: (8/3) * (-9/7) = -72/21 = -24/7.

38:29 Calculating Tangent of Half Angle (θ/2)

Formula: tan(θ/2) = ±√((1 - cos(θ))/(1 + cos(θ))).
Substitute cos(θ) = 3/5: ±√((1 - 3/5)/(1 + 3/5)) = ±√((2/5)/(8/5)) = ±√(2/8) = ±√(1/4) = ±1/2.
Determine sign: θ/2 is in Q1, so tan(θ/2) is positive.

42:21 Example with Quadrant 3: Cosecant and Cosine Negative

Given csc(θ) = -√5, implies sin(θ) = -1/√5 = -√5/5.
Given cos(θ) < 0.
Both sin(θ) and cos(θ) negative implies θ is in Quadrant 3 (π to 3π/2).

47:27 Finding Sine and Cosine in Quadrant 3

Using sin(θ) = y/r = -1/√5, let y=-1, r=√5.
Using Pythagorean theorem: x² + (-1)² = (√5)², x² + 1 = 5, x² = 4.
Since θ is in Q3, x must be negative: x = -2.
cos(θ) = x/r = -2/√5 = -2√5/5, tan(θ) = y/x = -1/-2 = 1/2.

53:23 Double Angle Calculations in Quadrant 3

sin(2θ) = 2sin(θ)cos(θ) = 2 * (-√5/5) * (-2√5/5) = 2 * (2*5)/25 = 20/25 = 4/5.
cos(2θ) = cos²θ - sin²θ = (-2√5/5)² - (-√5/5)² = (20/25) - (5/25) = 15/25 = 3/5.
Alternative cos(2θ) = 1 - 2sin²θ = 1 - 2*(-√5/5)² = 1 - 2*(5/25) = 1 - 10/25 = 15/25 = 3/5.

1:01:44 Half-Angle Quadrant Determination for Q3 θ

If θ is in Q3 (π to 3π/2), then θ/2 is in Q2 (π/2 to 3π/4).
This means sin(θ/2) is positive, cos(θ/2) is negative, tan(θ/2) is negative.

1:03:57 Calculating Half-Angle Formulas in Quadrant 2 (for θ/2)

sin(θ/2) = +√((1 - cos(θ))/2) = √((1 - (-2√5/5))/2) = √((5+2√5)/10).
cos(θ/2) = -√((1 + cos(θ))/2) = -√((1 + (-2√5/5))/2) = -√((5-2√5)/10).
tan(2θ) = 2tan(θ) / (1 - tan²(θ)) = 2(1/2) / (1 - (1/2)²) = 1 / (1 - 1/4) = 1 / (3/4) = 4/3.

1:16:59 Calculating Half-Angle Tangent (θ/2) in Quadrant 2

Formula: tan(θ/2) = ±√((1 - cos(θ))/(1 + cos(θ))).
Substitute cos(θ) = -2√5/5: ±√((1 + 2√5/5)/(1 - 2√5/5)) = ±√((5+2√5)/(5-2√5)).
Determine sign: θ/2 is in Q2, tan(θ/2) is negative.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Professor Leonard.