Trigonometry

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Trigonometry — Key Facts for Kenyatta University Basic ratios: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = sin/cos ASTC rule: All (QI), S (QII), T (QIII), C (QIV) Identities: sin²θ + cos²θ = 1; sec²θ = 1 + tan²θ; cosec²θ = 1 + cot²θ Compound angles: sin(A+B) = sin A cos B + cos A sin B; cos(A+B) = cos A cos B − sin A sin B ⚡ Exam tip: Kenyatta trig problems test compound angle formulas — substitution of standard angles speeds up solutions!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Trigonometry — Kenyatta University Study Guide

Six trig ratios:

• sin θ = 1/cosec θ
• cos θ = 1/sec θ
• tan θ = 1/cot θ = sin θ/cos θ

Standard angles:

• sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
• sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
• sin 60° = √3/2, cos 60° = ½, tan 60° = √3

Pythagorean identities:

• sin²θ + cos²θ = 1
• sec²θ = 1 + tan²θ
• cosec²θ = 1 + cot²θ

Compound angles:

• 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)

Double angle:

• sin 2θ = 2 sin θ cos θ
• cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ
• tan 2θ = 2 tan θ/(1 − tan²θ)

Sum-to-product:

• sin C + sin D = 2 sin[(C+D)/2] cos[(C−D)/2]
• cos C + cos D = 2 cos[(C+D)/2] cos[(C−D)/2]

General solution:

• sin θ = sin α → θ = nπ + (−1)^n α

• cos θ = cos α → θ = 2nπ ± α

• tan θ = tan α → θ = nπ + α

• Key formula: sin²θ + cos²θ = 1; sin(A+B) = sin A cos B + cos A sin B

• Common trap: tan(90°) is undefined — not infinity

• Exam weight: 2–3 questions per exam

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Trigonometry — Comprehensive Kenyatta Notes

Triple angle:

• sin 3θ = 3 sin θ − 4 sin³θ
• cos 3θ = 4 cos³θ − 3 cos θ
• tan 3θ = (3 tan θ − tan³θ)/(1 − 3 tan²θ)

Half-angle:

• sin(θ/2) = ±√[(1 − cos θ)/2]
• cos(θ/2) = ±√[(1 + cos θ)/2]
• tan(θ/2) = (1 − cos θ)/sin θ

Product-to-sum:

• sin A cos B = ½[sin(A+B) + sin(A−B)]
• cos A cos B = ½[cos(A+B) + cos(A−B)]
• sin A sin B = ½[cos(A−B) − cos(A+B)]

R format: a sin x + b cos x = R sin(x + α) where R = √(a² + b²) Maximum = R, minimum = −R

Conditional identities (A + B + C = π):

• sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
• tan A + tan B + tan C = tan A tan B tan C

Inverse trig basics:

• sin⁻¹x range: [−π/2, π/2]
• cos⁻¹x range: [0, π]
• tan⁻¹x range: (−π/2, π/2)
• sin⁻¹x + cos⁻¹x = π/2

Weierstrass substitution: t = tan(x/2): sin x = 2t/(1+t²), cos x = (1−t²)/(1+t²)

Trigonometric equations:

1. Express in terms of single trig function
2. Use inverse to find principal value
3. Write general solution

Height and distance:

• Angle of elevation: from horizontal up to object
• tan θ = height/distance

Radians: 180° = π rad; conversion: degrees × π/180

• Remember: ASTC for signs; always check quadrant for general solution
• Previous years: “Find tan 15°” [2023 KU]; “Solve sin x = cos x” [2024 KU]; “Prove sin(A+B) = sin A cos B + cos A sin B” [2024 KU]

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📊 Kenyatta University Exam Essentials

Detail	Value
Questions	50 (depending on course)
Time	2–3 hours
Marks	100
Format	Mix of short answer and problem solving

💡 Pro Tips

• For general solutions, remember periodic nature: sin, cos have period 2π; tan has period π
• When solving equations, convert to sin/cos, then solve
• Always check quadrant when finding arg values
• R format: a sin x + b cos x = R sin(x + α) gives max = R

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