Trigonometry: Ratios and Graphs

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your NECO exam.

The Three Primary Trigonometric Ratios:

In a right-angled triangle with angle $\theta$: $$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}, \quad \tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sin\theta}{\cos\theta}$$

The Unit Circle:

For a point $(x, y)$ on the unit circle making angle $\theta$ with the positive x-axis: $$\sin\theta = y, \quad \cos\theta = x, \quad \tan\theta = \frac{y}{x}$$

Key Values to Memorise:

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

⚡ NECO Tip: SOH-CAH-TOA helps you remember: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for NECO Mathematics students with a few days to months.

Pythagorean Identity: $$\sin^2\theta + \cos^2\theta = 1$$

From this: $1 + \tan^2\theta = \sec^2\theta$; $1 + \cot^2\theta = \csc^2\theta$

Radians: $$180° = \pi \text{ radians} \quad \Rightarrow \quad 1° = \frac{\pi}{180} \text{ rad}, \quad 1 \text{ rad} = \frac{180°}{\pi}$$

Area of a Triangle: $$\text{Area} = \frac{1}{2}ab\sin C \quad \text{(where } a \text{ and } b \text{ are two sides, } C \text{ is the included angle)}$$

Sine Rule: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Used when you know: (1) two angles and one side, or (2) two sides and an angle opposite one of them.

Cosine Rule: $$a^2 = b^2 + c^2 - 2bc\cos A$$ $$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

Used when you know: (1) two sides and the included angle, or (2) all three sides.

Trigonometric Graphs:

$y = \sin x$:

• Amplitude: 1
• Period: $360°$ ($2\pi$ radians)
• Domain: all real $x$; Range: $[-1, 1]$
• Passes through origin

$y = \cos x$:

• Amplitude: 1
• Period: $360°$
• Domain: all real $x$; Range: $[-1, 1]$
• Passes through $(0, 1)$

$y = \tan x$:

• Amplitude: not defined (unbounded)
• Period: $180°$
• Domain: all real $x$ except $90°, 270°…$; Range: all real numbers
• Vertical asymptotes at $x = 90° + 180°k$

⚡ NECO Common Mistakes:

• Mixing up when to use sine rule vs cosine rule
• Forgetting to use the angle opposite the known side in the sine rule
• In ambiguous case of sine rule (SSA), there may be two possible triangles — solve for the second angle ($180° - \theta$)
• Measuring angles in degrees when the formula requires radians or vice versa

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for NECO and JAMB Mathematics preparation.

Compound Angle Formulae:

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

Double Angle Formulae:

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

Transformations of Trigonometric Graphs:

• $y = a\sin x$: amplitude changes to $|a|$
• $y = \sin(bx)$: period changes to $\frac{360°}{|b|}$
• $y = \sin x + c$: vertical shift, period unchanged
• $y = \sin(x - c)$: horizontal shift (phase shift)

Small Angle Approximations (for $\theta$ in radians): When $\theta$ is very small (close to 0): $$\sin\theta \approx \theta, \quad \cos\theta \approx 1 - \frac{\theta^2}{2}, \quad \tan\theta \approx \theta$$

Trigonometric Identities to Prove:

Example: Prove $\sin^4 x - \cos^4 x = \sin^2 x - \cos^2 x$ LHS: $(\sin^2 x - \cos^2 x)(\sin^2 x + \cos^2 x) = (\sin^2 x - \cos^2 x)(1) = \sin^2 x - \cos^2 x = $ RHS ✓

Solving Trigonometric Equations:

Example: Solve $\sin x = \frac{1}{2}$ for $0° \leq x \leq 360°$: Reference angle = $30°$. $\sin x$ is positive in Q1 and Q2. $x = 30°$ or $x = 180° - 30° = 150°$

Example: Solve $2\cos^2 x - 1 = 0$: $2\cos^2 x = 1 \Rightarrow \cos^2 x = \frac{1}{2} \Rightarrow \cos x = \pm\frac{1}{\sqrt{2}}$ $x = 45°, 135°, 225°, 315°$

Graphs of $y = a\sin(bx + c)$:

• Amplitude: $|a|$
• Period: $\frac{360°}{|b|}$
• Phase shift: $-\frac{c}{b}$ (positive $c$ shifts left, negative $c$ shifts right)

NECO/JAMB Patterns:

• NECO frequently asks: solve triangles using sine and cosine rules; prove trigonometric identities; solve trigonometric equations within a given range; sketch and describe transformations of trigonometric graphs; find exact values using special angles

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