Trigonometry Review

Trigonometry Review & Upcoming Class

• The last class on Trigonometry is on Saturday
• An offer will be available one day prior to the last class

Question 1

• An angle is divided into two parts: X and α.
• tan(X) = λ * tan(α).
• Objective is to find the value of sin(X) / sin(α).

Question 1: Options

• Option A: λ / (λ + 1)
• Option B: (λ - 1) / λ
• Option C: (λ - 1) / (λ + 1)
• Option D: Incorrect Value

Important Note on Tangents

• "Tangents" refers to tan(θ).
• "Tangent of theta" is the origin of the term "tan θ."

Question 2

• Simplify the trigonometric expression: (sin 5x - sin 3x) / (cos 5x + 5 cos 3x + 10 cos x)

Question 2: Options

• Option A: cos 2x
• Option B: 2 cos x
• Option C: cos² x
• Option D: 1 + cos x

Warm-up Question (KVPY Exam)

• Given sin θ = (a² - b²) / (a² + b²).
• Also given a * sin θ / cos² θ = b * cos θ / sin² θ = 0.
• Find the value of (aX)^(2/3) + (b/X)^(2/3).

Strategy for Warm-up Question

• Eliminate cos θ and sin θ to solve for the unknown expression.
• Retain 'a', 'b', and 'X' in the equations.
• Eliminate trigonometric functions.

Initial Steps for Warm-up Question

• Start with the second equation.
• Replace cos θ with C and sin θ with S for simplicity (only when single angles)

Transforming the Second Equation

• Rewrite the second equation as: aX * (S/C²) = b * (C/S²).
• Rearrange to get X/b = (C³/S³).

Connecting to Tangent

• Express C³/S³ as (cos θ / sin θ)³ = cot³ θ.
• Therefore, X/b = cot³ θ.

Isolating Cotangent

• cot θ = (aX / b)^(1/3).

Utilizing the First Equation

• Eliminate cos θ and sin θ.
• cot θ is the ratio of base to perpendicular (B/P).
• Base/Perpendicular = (AX/B)^(1/3).

Determining Hypotenuse

• Base = (aX)^(1/3).
• Perpendicular = b^(1/3)
• Hypotenuse = √((aX)^(2/3) + b^(2/3)).

Calculating cos θ and sin θ

• cos θ = (aX)^(1/3) / √((aX)^(2/3) + b^(2/3)).
• sin θ = b^(1/3) / √((aX)^(2/3) + b^(2/3)).

Substituting into First Equation

• Substitute cos θ and sin θ values into the first equation: sin θ = (a² - b²) / (a² + b²).
• b^(2/3) / √((aX)^(2/3) + b^(2/3)) = (a² - b²) / (a² + b²).

Solving for the Expression

• Simplifies to (aX)^(2/3) + b^(2/3) = (a² - b²)^(2/3).
• (aX)^(2/3) + (b/X)^(2/3) = (a² - b²)^(2/3)

Formula 8: Cosine Triple Angle

• Deriving the formula for cos 3x.
• Split 3x as x + 2x.

Applying Cosine Addition Formula

• cos 3x = cos(x + 2x) = cos x * cos 2x - sin x * sin 2x.

Converting to Cosine Terms

• Using cos 2x = 2 cos² x - 1.
• sin 2x = 2 sin x cos x.
• cos 3x = cos x (2 cos² x - 1) - sin x (2 sin x cos x few years).

Simplifying and Substituting

• Substitute sin² x with 1 - cos² x.
• Simplify to get: 2 cos³ x - cos x - 2(1 - cos² x) cos x.
• Further simplification leads to: 4 cos³ x - 3 cos x.

Final Formula for cos 3x

• cos 3x = 4 cos³ x - 3 cos x.

Formula 9: Sine Triple Angle

• Deriving the formula for sin 3x.
• Objective: Express sin 3x in terms of sine functions exclusively.

Formulas for Trigonometric Functions

• Focuses on trigonometric formulas for multiple angles and their derivations.
• Aims to provide methods to remember them effectively.
• Discusses methods for deriving and remembering trigonometric formulas, especially for multiple angles.

Sine of 3x (sin 3x)

• Formula derived: sin 3x = 3sin x - 4sin³x
• cos 3x = 4cos³x - 3sin x (inverse of sin 3x formula with cosine).

Tangent of x + y (tan (x+y))

• Formula for tangent of sum of angles: tan(x+y) = (tan x + tan y) / (1 - tan x tan y)
• Obtained by dividing numerator and denominator of sin(x+y) / cos(x+y) by cos x cos y.
• Tangent of difference of angles: tan(x-y) = (tan x - tan y) / (1 + tan x tan y)

Tangent of 3x (tan 3x)

• Derivation uses the formula for tan(x+y), letting y = 2x.
• The derived formula of tan 3x formula simplifies to (3 tan x - tan³x) / (1 - 3tan²x).

Tangent of x + y + z (tan (x+y+z))

• Formula: tan(x+y+z) = (tan x + tan y + tan z - tan x tan y tan z) / (1 - tan x tan y - tan y tan z - tan z tan x).

General formula for tangent of multiple angles

• Expressed as a ratio of sums of products of tangents: (s1 - s3 + s5 - ...) / (1 - s2 + s4 - ...), where sn is the sum of products of tangents taken n at a time.
• s2 = sum of tangents of angles taken two at a time.
• s3 = the sum of the tangents of angles taken three at a time

Product-to-Sum Formulas

• Used the sum and difference formulas for sine and cosine to derive product-to-sum formulas.
• sin(x+y) + sin(x-y) = 2 sin x cos y
• sin(x+y) - sin(x-y) = 2 cos x sin y
• cos(x+y) + cos(x-y) = 2 cos x cos y
• cos(x+y) - cos(x-y) = -2 sin x sin y

Sum-to-Product (CD) Formulas

• Introduces substitution C = x + y and D = x - y to convert product-to-sum formulas into sum-to-product formulas.
• sin C + sin D = 2 sin((C+D)/2) cos((C-D)/2)
• sin C - sin D = 2 cos((C+D)/2) sin((C-D)/2)
• cos C + cos D = 2 cos((C+D)/2) cos((C-D)/2)
• cos C - cos D = -2 sin((C+D)/2) sin((C-D)/2)

Additional Trigonometric Formulas

• cos(A - B) * cos(A + B) = cos²A - sin²B
• sin(A + B) * sin(A - B) = sin²A - sin²B

Half-Angle Formulas

• Focuses on expressing trigonometric functions in terms of half angles; for example, cos θ in terms of θ/2.
• Cos(θ) related to half-angle: Derived from Cos(2θ) formulas (1-2sin²(θ), 2cos²(θ)-1, etc) replacing θ with θ/2:
• cos θ = 1 - 2sin²(θ/2)
• cos θ = 2cos²(θ/2) - 1
• tan² θ = (1 - cos 2θ) / (1 + cos 2θ) is transformed to tan²(θ/2) = (1 - cos θ) / (1 + cos θ).

Trigonometric Ratios for Specific Angles

• Focuses on determining trigonometric ratios for standard angles such as 15°, 75°, and 22.5°.

Sine of 15 degrees (sin 15°)

• sin 15° = sin(45° - 30°) = (√3 - 1) / (2√2); equals cos 75°

Sine of 75 degrees (sin 75°)

• sin 75° = sin(45° + 30°) = (√3 + 1) / (2√2); equals cos 15°

Tangent of 15 degrees (tan 15°)

• tan 15° = sin 15° / cos 15° = (√3 - 1) / (√3 + 1)

Tangent of 75 degrees (tan 75°)

• tan 75° = 1 / cot 75° = (√3 + 1) / (√3 - 1)

Sine of 22.5 degrees (sin 22.5°)

• Obtained by using the formula cos 45° = 1 - 2sin²(22.5°).
• sin 22.5° = √(2 - √2) / 2;

Cosine of 22.5 degrees (cos 22.5°)

• Derived from the formula cos 45° = 2cos²(22.5°) - 1.
• Simplified, cos 22.5° = √(2 + √2) / 2; equals sin 67.5°

Tangent of 22.5 degrees (tan 22.5°)

• Derivation: tan 22.5° = sin 22.5° / cos 22.5° = √(√2 - 1) / √(√2 + 1)
• tan 22.5° also is cot 67.5°

Tangent of 67.5 degrees (tan 67.5°)

• tan 67.5° = 1 / cot 67.5° = √(√2 + 1) / √(√2 - 1); which is also cot 22.5°