Trigonometry

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Trigonometry — Key Facts for BUET Basic ratios: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = sin/cos ASTC rule: All ( QI), S (QII), T (QIII), C (QIV) for positive values 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: BUET trig problems test compound angle formulas and solving equations — substitution of standard angles is a fast approach!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Trigonometry — BUET Study Guide

Six trig ratios and reciprocals:

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

Standard angles to memorise:

• 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
• sin 0° = 0, cos 0° = 1, tan 0° = 0
• sin 90° = 1, cos 90° = 0, tan 90° = undefined

Pythagorean identities:

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

Compound angle 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)

Double angle formulas:

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

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]

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)]

General solution of equations:

• sin θ = sin α → θ = nπ + (−1)^n α
• cos θ = cos α → θ = 2nπ ± α
• tan θ = tan α → θ = nπ + α

Radians:

• 180° = π rad

• Conversion: degrees × π/180 = radians

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

• Common trap: tan(90°) is undefined — not infinity; always be careful with domain restrictions

• Exam weight: 2–3 questions per exam (8–12 marks); foundational for other topics

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Trigonometry — Comprehensive BUET Notes

Triple angle formulas:

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

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

• sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C
• cos A + cos B + cos C = 1 + 4 sin(A/2) sin(B/2) sin(C/2)
• tan A + tan B + tan C = tan A tan B tan C

Maximum-minimum values: For a sin x + b cos x: max = √(a² + b²), min = −√(a² + b²) Achieved at x where tan x = b/a (shifted)

Half-angle formulas:

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

Weierstrass substitution (t = tan(x/2)):

• sin x = 2t/(1+t²)
• cos x = (1−t²)/(1+t²)
• tan x = 2t/(1−t²) Used for solving trigonometric equations and integrals

R format for a sin x + b cos x: a sin x + b cos x = R sin(x + α) where R = √(a² + b²) and α = tan⁻¹(b/a)

Proving trigonometric identities:

• Always start from more complex side
• Use basic identities to simplify
• Convert to sin/cos if needed

Inverse trigonometric 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 for x ∈ [−1, 1]
• tan⁻¹x + cot⁻¹x = π/2 for x ∈ ℝ

Radians vs degrees: sin x / x → 1 as x → 0 only when x is in radians Always use radians in limit calculations

Altitude and angle problems:

• sin θ = opposite/hypotenuse
• In right triangles: if angle is θ, opposite side = hyp × sin θ, adjacent = hyp × cos θ
• tan θ = opposite/adjacent

Transformation cascade: sin 4x = 2 sin 2x cos 2x = 4 sin x cos x (1 − 2 sin² x)

Nth roots and trigonometric form: For complex number in trig form: z = r(cos θ + i sin θ) z^n = r^n (cos nθ + i sin nθ)

Trigonometric series:

• Sum of sines: S = sin a + sin(a+d) + … + sin(a+(n-1)d)
• Use sum-to-product on pairs

Identities for specific angles:

• tan 15° = 2 − √3

• tan 75° = 2 + √3

• sin 15° = (√6 − √2)/4

• cos 15° = (√6 + √2)/4

• Remember: ASTC for signs; sin² + cos² = 1; always check quadrant when finding arg(sin/coss value); general solution for sin θ = sin α: θ = nπ + (−1)^n α

• Previous years: “Find value of tan 15°” [2023 BUET]; “Solve sin 2x = cos x for x ∈ [0, 2π]” [2024 BUET]; “Prove sin(A+B) = sin A cos B + cos A sin B” [2024 BUET]

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📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Mathematics

• Calculus (Differentiation + Integration) — highest weight
• Algebra (Quadratics, AP/GP/HP) — very high weight
• Coordinate Geometry (Circle, Conics) — high weight
• Trigonometry — medium-high weight
• Complex Numbers — medium weight

📝 Previous Year Question Patterns

• Trigonometry: 2–3 questions per exam, 8–12 marks
• Common patterns: compound angle formulas, solving trig equations, proving identities, finding specific angle values
• Weight: medium-high — foundations needed for calculus

💡 Pro Tips

• Always use radians when doing calculus involving trig functions
• For general solutions, remember the periodic nature: sin and cos have period 2π, tan has period π
• When solving trig equations, express everything in terms of sin and cos, then substitute
• For maximum of a sin x + b cos x, always convert to R sin(x + α) form
• Remember tan(A+B) formula sign — it’s (tan A + tan B)/(1 − tan A tan B), not addition in denominator

🔗 Official Resources

• BUET Official
• BUET Admission Portal

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