Trigonometry Tricks for Exams — Edwin Ogie Library

Trigonometry Tricks for Exams — Cheat-sheet & Practice

Quick identities, angle facts, speed tips, worked examples, exercises and a 40‑question JAMB‑style CBT (15 mins)

Edwin Ogie Library

Cheat-sheet: Core Identities & Special Angles

Basic definitions

For a right triangle: sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj. Also tanθ = sinθ/cosθ.

Pythagorean identities

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Angle sum & difference

• sin(α±β) = sinα cosβ ± cosα sinβ
• cos(α±β) = cosα cosβ ∓ sinα sinβ
• tan(α±β) = (tanα ± tanβ)/(1 ∓ tanα tanβ)

Double & half angles

• sin2θ = 2sinθ cosθ
• cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
• tan(θ/2) = ±√((1−cosθ)/(1+cosθ))

Reciprocal & cofunction

• cscθ = 1/sinθ, secθ = 1/cosθ, cotθ = 1/tanθ
• sin(90°−θ)=cosθ, cos(90°−θ)=sinθ, tan(90°−θ)=cotθ

Product-sum

• sinα sinβ = 1/2[cos(α−β) − cos(α+β)]
• cosα cosβ = 1/2[cos(α−β) + cos(α+β)]

Special angle values (degrees)

θ	0°	30°	45°	60°	90°
sinθ	0	1/2	√2/2	√3/2	1
cosθ	1	√3/2	√2/2	1/2	0
tanθ	0	1/√3	1	√3	—

Unit circle (useful for quick angle values)

Speed Tips & Exam Shortcuts

• Memorize sin, cos, tan for 0°,30°,45°,60°,90° — these appear often.
• Use cofunction symmetry: change sin(90°−θ) to cosθ to remove negative signs quickly.
• Reduce larger angles: sin(360°−θ)= −sinθ, cos(360°−θ)=cosθ.
• When given tanθ value, compute sinθ or cosθ using triangle identities for speed.
• Replace sum/difference forms with known values for 30°,45°,60° to avoid heavy algebra.
• Recognize patterns: sin²θ often pairs with 1−cos²θ and simplifies quickly.
• When stuck, test special angles (0°,30°,45°,60°,90°) to eliminate wrong MCQ options fast.

Worked Examples (quick)

1. Example 1: Evaluate sin75° quickly.
   Use sum: sin(45°+30°)=sin45 cos30 + cos45 sin30 = (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6+√2)/4
2. Example 2: Simplify 1−sin²θ → cos²θ (Pythagorean), useful in MCQs.
3. Example 3: If tanθ = 3/4 and θ in QI, find sinθ. Construct triangle: opp=3, adj=4, hyp=5 → sinθ = 3/5.

Practice Exercises (Short)

1. Find cos(75°) using angle sum/difference.
2. Simplify tan(45°+θ).
3. Given sinθ = 3/5 (θ in QI), find tanθ.
4. Prove 1+tan²θ = sec²θ.
5. Evaluate sin(−30°) and cos(−30°).

Answers at the bottom of this note — scroll or use the page find.

40‑Question JAMB‑Style CBT — Time: 15 minutes

Instructions: The timer starts after a short 5‑second countdown that begins when you click Start Quiz. You have 15 minutes (900 seconds) to answer 40 multiple-choice questions. Submit before time runs out. This simulates exam timing — good practice for JAMB.

Time: 15:00

Question: 0/40

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Created by Edwin Ogie — Trigonometry tricks, cheat‑sheet and 40‑question CBT. Contact: edwinogielibrary@gmail.com

Practice Answers

1. cos75° = cos(45°+30°) = cos45 cos30 − sin45 sin30 = (√2/2)(√3/2) − (√2/2)(1/2) = (√6 − √2)/4
2. tan(45°+θ) = (tan45 + tanθ)/(1 − tan45 tanθ) = (1 + tanθ)/(1 − tanθ)
3. If sinθ = 3/5 then cosθ = 4/5, so tanθ = (3/5)/(4/5) = 3/4
4. 1+tan²θ = sec²θ is derived from dividing sin²θ+cos²θ=1 by cos²θ.
5. sin(−30°) = −1/2; cos(−30°) = √3/2