TRIGONOMETRY

Complete Trigonometry Notes (Unit 5)

5.0 Review: Compound Angles & Standard Values

Compound Angles: For angles A and B:

• Addition/Subtraction Formulas:
  • sin(A ± B) = sin A cos B ± cos 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)
  • cot(A ± B) = (cot A cot B ∓ 1)/(cot B ± cot A)

5.1 Multiple Angles

Multiple Angles: 2A, 3A, etc.

For 2A:

• sin 2A = 2 sin A cos A
• cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A
• tan 2A = 2 tan A / (1 - tan² A)
• cot 2A = (cot² A - 1)/(2 cot A)

Alternative Forms:

• sin 2A = 2 tan A / (1 + tan² A)
• cos 2A = (1 - tan² A)/(1 + tan² A)
• 2 cos² A = 1 + cos 2A → cos² A = (1 + cos 2A)/2
• 2 sin² A = 1 - cos 2A → sin² A = (1 - cos 2A)/2
• tan² A = (1 - cos 2A)/(1 + cos 2A)
• cot² A = (1 + cos 2A)/(1 - cos 2A)

Geometrical Proof: Unit circle with diameter PR, point Q, angle A at periphery → central 2A.

For 3A:

• sin 3A = 3 sin A - 4 sin³ A
• cos 3A = 4 cos³ A - 3 cos A
• tan 3A = (3 tan A - tan³ A)/(1 - 3 tan² A)
• Derived from sin(2A + A), cos(2A + A).

Common Identities & Proofs:

• cot A = ± √[(1 + cos 2A)/(1 - cos 2A)]
• sin 2A / (1 + cos 2A) = tan A
• (1 - cos 2A + sin 2A)/(1 + cos 2A + sin 2A) = tan A
• tan(π/4 + θ) = cos 2θ / (1 - sin 2θ)
• √2 + √(2 + √(2 + 2 cos 8θ)) = 2 cos θ
• cos⁶ θ + sin⁶ θ = (1/4)(1 + 3 cos² 2θ)
• Many with 45° ± θ, powers, products.

Special Values (from multiple angles):

• sin 18° = (√5 - 1)/4
• cos 18° = √(10 + 2√5)/4
• sin 36° = √(10 - 2√5)/4
• cos 36° = (√5 + 1)/4
• tan 18° = √(25 - 10√5)/5 or 2 - √5 (rationalized)

5.2 Submultiple Angles (A/2, A/3)

Submultiple: A/2, A/3, etc.

For A/2:

• sin A = 2 sin(A/2) cos(A/2)
• cos A = cos²(A/2) - sin²(A/2) = 2 cos²(A/2) - 1 = 1 - 2 sin²(A/2)
• tan A = 2 tan(A/2)/(1 - tan²(A/2))
• sin A = 2 tan(A/2)/(1 + tan²(A/2))
• cos A = (1 - tan²(A/2))/(1 + tan²(A/2))

For A/3:

• sin A = 3 sin(A/3) - 4 sin³(A/3)
• cos A = 4 cos³(A/3) - 3 cos(A/3)
• tan A = (3 tan(A/3) - tan³(A/3))/(1 - 3 tan²(A/3))

Identities:

• sin θ / (1 + cos θ) = tan(θ/2)
• (1 - cos θ)/sin θ = cot(θ/2)
• 1 + sin θ = (sin(θ/2) + cos(θ/2))²
• 1 - sin θ = (sin(θ/2) - cos(θ/2))²

Comparative Table (Multiple vs Submultiple): As in text (20 pairs mirrored).

5.3 Transformation Formulas

Product to Sum:

• 2 sin A cos B = sin(A+B) + sin(A-B)
• 2 cos A sin B = sin(A+B) - sin(A-B)
• 2 cos A cos B = cos(A+B) + cos(A-B)
• 2 sin A sin B = cos(A-B) - cos(A+B)

Sum to Product:

• 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)

Applications: Simplify products/sums, prove identities (e.g., sin θ sin(60°±θ) = (1/4) sin 3θ, special products like sin 20° sin 40° sin 80° = √3/16).

5.4 Conditional Identities (A + B + C = 180°)

• tan A + tan B + tan C = tan A tan B tan C
• sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
• cos 2A + cos 2B + cos 2C = -1 - 2 cos 2A cos 2B cos 2C (variants)
• sin A + sin B + sin C = 4 cos(A/2) cos(B/2) cos(C/2)
• Many half-angle, product forms.

5.5 Trigonometric Equations

• Definition: Equation with trig ratios of unknown angle.
• Solution: Values satisfying equation.
• CAST Rule: Quadrants signs (All, Sin, Tan, Cos positive).
• Steps:
  • Sign → quadrant.
  • Reference angle.
  • Adjust: 180° - θ (2nd), 180° + θ (3rd), 360° - θ (4th).
  • General: +360°k.
• Range: sin/cos [-1,1].

5.6 Height and Distance

• Elevation: Angle above horizontal to higher object.
• Depression: Angle below horizontal to lower object.
• Applications: tan θ = height / distance; change in angle/shadow for multiple measurements.

5 Most Common Questions

1. If sin A = 3/5, find sin 2A, cos 2A, tan 2A.
2. Prove sin 3A = 3 sin A - 4 sin³ A.
3. Solve sin x + cos x = √2 (0° ≤ x ≤ 360°).
4. If A + B + C = 180°, prove sin A + sin B + sin C = 4 cos(A/2) cos(B/2) cos(C/2).
5. A tower casts a shadow 40 m longer at 30° sun elevation than at 60°. Find height.

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