“Trigonometry: Ratios and Identities”

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Trigonometry: Ratios and Identities — Quick Facts for JAMB

Basic Trigonometric Ratios (in a right triangle): For angle $\theta$:

• $\sin\theta = \frac{opposite}{hypotenuse}$
• $\cos\theta = \frac{adjacent}{hypotenuse}$
• $\tan\theta = \frac{opposite}{adjacent} = \frac{\sin\theta}{\cos\theta}$

Reciprocal Identities:

• $\csc\theta = 1/\sin\theta$
• $\sec\theta = 1/\cos\theta$
• $\cot\theta = 1/\tan\theta = \cos\theta/\sin\theta$

Pythagorean Identities:

• $\sin^2\theta + \cos^2\theta = 1$
• $1 + \tan^2\theta = \sec^2\theta$
• $1 + \cot^2\theta = \csc^2\theta$

Values to Memorise:

Angle	$\sin$	$\cos$	$\tan$
0°	0	1	0
30°	1/2	$\sqrt{3}/2$	$1/\sqrt{3}$
45°	$\sqrt{2}/2$	$\sqrt{2}/2$	1
60°	$\sqrt{3}/2$	1/2	$\sqrt{3}$
90°	1	0	undefined

⚡ Exam tip: For angles beyond 90°, use ASTC (All Students Take Calculus): All functions positive in QI, only Sin positive in QII, only Tan positive in QIII, only Cos positive in QIV.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Trigonometry: Ratios and Identities — JAMB UTME Study Guide

Angle Sum and Difference:

• $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$
• $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
• $\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

Double Angle:

• $\sin 2A = 2\sin A \cos A$
• $\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$
• $\tan 2A = \frac{2\tan A}{1 - \tan^2 A}$

Half Angle:

• $\sin(A/2) = \pm\sqrt{\frac{1-\cos A}{2}}$
• $\cos(A/2) = \pm\sqrt{\frac{1+\cos A}{2}}$ (Signs depend on the quadrant of $A/2$)

Solving Trigonometric Equations: Example: Solve $\sin x = \sqrt{3}/2$ for $0° \leq x \leq 360°$. $\sin x = \sqrt{3}/2$ means $x = 60°$ or $180° - 60° = 120°$. Answers: $60°, 120°$.

Example: Solve $2\cos^2 x - 1 = 0$ for $0° \leq x \leq 360°$. $\cos 2x = 0$. So $2x = 90°, 270°, 450°, 630°…$ In range: $2x = 90°, 270°$. So $x = 45°, 135°$.

Secant, Cosecant, Cotangent: Graphs and ranges:

• $\sec\theta$: $\sec\theta \geq 1$ or $\sec\theta \leq -1$
• $\csc\theta$: $\csc\theta \geq 1$ or $\csc\theta \leq -1$
• $\cot\theta$: all real values

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Trigonometry: Ratios and Identities — Comprehensive Mathematics Notes

Derivation of Key Identities:

From Pythagorean theorem in a right triangle: $a^2 + b^2 = c^2$. Dividing by $c^2$: $(a/c)^2 + (b/c)^2 = 1$. So $\sin^2\theta + \cos^2\theta = 1$.

Dividing $\sin^2\theta + \cos^2\theta = 1$ by $\cos^2\theta$: $\tan^2\theta + 1 = \sec^2\theta$.

Dividing by $\sin^2\theta$: $1 + \cot^2\theta = \csc^2\theta$.

Radians vs Degrees: $180° = \pi$ radians. Conversion: angle in radians = angle in degrees × $\pi/180$. Arc length $s = r\theta$ where $\theta$ is in radians. Area of sector = $\frac{1}{2}r^2\theta$.

Example: Arc length of a circle of radius 5 cm with angle $60°$: $s = 5 \times (\pi/3) = 5\pi/3$ cm.

The Law of Cosines: For any triangle with sides $a, b, c$ and opposite angles $A, B, C$: $a^2 = b^2 + c^2 - 2bc\cos A$ $b^2 = a^2 + c^2 - 2ac\cos B$ $c^2 = a^2 + b^2 - 2ab\cos C$

This is a generalisation of Pythagoras. When $C = 90°$, $\cos C = 0$, and we get $c^2 = a^2 + b^2$.

The Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ where $R$ = circumradius.

Area of a Triangle: Using two sides and included angle: $\Delta = \frac{1}{2}ab\sin C$.

Product-to-Sum Identities:

• $\sin A \cos B = \frac{1}{2}[\sin(A+B) + \sin(A-B)]$
• $\cos A \sin B = \frac{1}{2}[\sin(A+B) - \sin(A-B)]$
• $\cos A \cos B = \frac{1}{2}[\cos(A+B) + \cos(A-B)]$
• $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

Sum-to-Product Identities:

• $\sin A + \sin B = 2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$
• $\sin A - \sin B = 2\cos\frac{A+B}{2}\sin\frac{A-B}{2}$
• $\cos A + \cos B = 2\cos\frac{A+B}{2}\cos\frac{A-B}{2}$
• $\cos A - \cos B = -2\sin\frac{A+B}{2}\sin\frac{A-B}{2}$

Trigonometric Equations — Advanced:

Example: Solve $\sin 2x = \cos x$ for $0 \leq x < 2\pi$. $\sin 2x = 2\sin x \cos x$. So $2\sin x \cos x = \cos x$. $2\sin x \cos x - \cos x = 0$. $\cos x(2\sin x - 1) = 0$. Case 1: $\cos x = 0$ → $x = \pi/2, 3\pi/2$. Case 2: $\sin x = 1/2$ → $x = \pi/6, 5\pi/6$. All solutions: $x = \pi/6, \pi/2, 5\pi/6, 3\pi/2$.

Inverse Trigonometric Functions:

• $\arcsin x$: domain $[-1,1]$, range $[-\pi/2, \pi/2]$
• $\arccos x$: domain $[-1,1]$, range $[0, \pi]$
• $\arctan x$: domain $\mathbb{R}$, range $(-\pi/2, \pi/2)$

$\arcsin(\sin x) = x$ only when $x \in [-\pi/2, \pi/2]$. Outside this range, the result is different.

JAMB Pattern Analysis: JAMB questions frequently test: (1) Using Pythagorean identities to simplify expressions, (2) Solving trig equations using standard values, (3) Angle sum/difference formulas, (4) Using $\sin^2\theta + \cos^2\theta = 1$ to find $\sin\theta$ or $\cos\theta$ given one of them, (5) Word problems involving heights and distances using $\tan\theta$ and $\sin\theta$. Common error: forgetting that $\tan 90°$ is undefined. JAMB 2022: “If $\sin\theta = 3/5$, find $\cos\theta$ and $\tan\theta$.” Answer: $\cos\theta = \sqrt{1 - 9/25} = 4/5$ (taking positive in QI), $\tan\theta = 3/4$.

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📊 JAMB Exam Essentials

Detail	Value
Questions	180 MCQs (UTME)
Subjects	4 subjects (language + 3 for course)
Time	2 hours
Marking	+1 per correct answer
Score	400 max (used for university admission)
Registration	January – February each year

🎯 High-Yield Topics for JAMB

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• Physics (Mechanics + Optics) — 35 marks
• Mathematics (Algebra + Geometry) — 40 marks

📝 Previous Year Question Patterns

• Q: “The process of photosynthesis requires…” [2024 Biology]
• Q: “The electronic configuration of Fe is…” [2024 Chemistry]
• Q: “Find the value of x if 2x + 5 = 15…” [2024 Mathematics]

💡 Pro Tips

• Use of English carries the most weight — master grammar rules and comprehension strategies
• JAMB syllabus is your Bible — questions come directly from it. Download and use it.
• Past questions are highly predictive — repeat patterns appear every year
• For Science students, Biology and Chemistry are high-scoring if you study NCERT-level content

🔗 Official Resources

• JAMB Official
• JAMB Syllabus

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