“Trigonometry: Ratios and Graphs”

🟢 Lite — Quick Review (1h–1d)

Rapid summary of trigonometry for NABTEB mathematics.

Trigonometry deals with the relationships between the sides and angles of triangles.

The Three Primary Ratios — SOH-CAH-TOA:

For a right-angled triangle with angle $\theta$:

Ratio	Formula	Meaning
Sine	$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$	SOH
Cosine	$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$	CAH
Tangent	$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$	TOA

The Hypotenuse is always the longest side, opposite the right angle.

Pythagoras’ Theorem: $$a^2 + b^2 = c^2$$ Where $c$ is the hypotenuse, $a$ and $b$ are the other two sides.

Standard Angles:

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

Reciprocal Ratios: $$\csc\theta = \frac{1}{\sin\theta} = \frac{\text{hypotenuse}}{\text{opposite}}$$ $$\sec\theta = \frac{1}{\cos\theta} = \frac{\text{hypotenuse}}{\text{adjacent}}$$ $$\cot\theta = \frac{1}{\tan\theta} = \frac{\text{adjacent}}{\text{opposite}}$$

⚡ NABTEB Exam Tip: In trigonometry questions, always identify which sides are opposite, adjacent, and hypotenuse relative to the given angle. Draw a diagram if one is not provided.

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🟡 Standard — Regular Study (2d–2mo)

For NABTEB students who want thorough understanding.

Sine and Cosine Rules:

Sine Rule (for any triangle): $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Use when you know two angles and one side, OR two sides and an angle opposite one of them.

Cosine Rule: $$a^2 = b^2 + c^2 - 2bc\cos A$$ $$b^2 = a^2 + c^2 - 2ac\cos B$$ $$c^2 = a^2 + b^2 - 2ab\cos C$$

Use when you know two sides and the included angle, OR all three sides.

Area of a Triangle: $$A = \frac{1}{2}ab\sin C$$

Where $a$ and $b$ are two sides and $C$ is the included angle.

Trigonometric Identities:

Fundamental identity: $$\sin^2\theta + \cos^2\theta = 1$$

From this: $$\tan\theta = \frac{\sin\theta}{\cos\theta}$$ $$\sin^2\theta = 1 - \cos^2\theta$$ $$\cos^2\theta = 1 - \sin^2\theta$$

Angles of Elevation and Depression:

Type	Description	How to measure
Angle of elevation	Angle ABOVE the horizontal	Look UP from object
Angle of depression	Angle BELOW the horizontal	Look DOWN from object

The angle of elevation equals the angle of depression when lines are parallel.

Solving Trigonometric Equations:

Example: Find $\theta$ if $\sin\theta = 0.5$ and $\theta$ is between 0° and 90°: $$\theta = 30°$$

If between 0° and 360°: $$\theta = 30° \text{ or } 150° \text{ (since } \sin 150° = \sin 30° = 0.5)$$

⚡ NABTEB Exam Tip: Remember that $\tan 90°$ is undefined (division by zero). In graphs of $y = \tan\theta$, there are vertical asymptotes at 90°, 270°, etc.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage of trigonometry for thorough NABTEB preparation.

The Unit Circle and Radian Measure:

One radian = angle subtended when arc length equals radius. $$\pi \text{ radians} = 180°$$ $$1° = \frac{\pi}{180} \text{ radians}$$ $$1 \text{ radian} = \frac{180}{\pi} ≈ 57.3°$$

Converting: $$30° = \frac{\pi}{6} \text{ rad}$$ $$45° = \frac{\pi}{4} \text{ rad}$$ $$60° = \frac{\pi}{3} \text{ rad}$$ $$90° = \frac{\pi}{2} \text{ rad}$$

Trigonometric Graphs:

$y = \sin x$:

• Amplitude: 1
• Period: 360° ($2\pi$ radians)
• Passes through origin
• Maximum at 90°, minimum at 270°
• Symmetric about 0°, 180°, 360°

$y = \cos x$:

• Amplitude: 1
• Period: 360° ($2\pi$ radians)
• Starts at maximum (1) when $x = 0$
• Minimum at 180°, maximum at 0° and 360°

$y = \tan x$:

• Amplitude: not defined
• Period: 180° ($\pi$ radians)
• Vertical asymptotes at 90°, 270°
• Passes through origin

Transformations:

Transformation	Effect on $y = \sin x$
$y = a\sin x$	Vertical stretch by factor $a$
$y = \sin bx$	Horizontal compression by factor $b$
$y = \sin(x + c)$	Horizontal shift left by $c$
$y = \sin x + d$	Vertical shift up by $d$

Compound Angle Formulas:

$$\sin(A + B) = \sin A \cos B + \cos A \sin B$$ $$\sin(A - B) = \sin A \cos B - \cos A \sin B$$ $$\cos(A + B) = \cos A \cos B - \sin A \sin B$$ $$\cos(A - B) = \cos A \cos B + \sin A \sin B$$

$$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}$$ $$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}$$

Double Angle Formulas:

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

Solving Triangles — Worked Examples:

Example 1 (Sine Rule): Given $a = 7$, $A = 30°$, $B = 50°$, find $b$: $$\frac{b}{\sin B} = \frac{a}{\sin A}$$ $$b = 7 \times \frac{\sin 50°}{\sin 30°} = 7 \times \frac{0.766}{0.5} = 10.72$$

Example 2 (Cosine Rule): Given $a = 8$, $b = 5$, $C = 60°$, find $c$: $$c^2 = 8^2 + 5^2 - 2(8)(5)\cos 60°$$ $$c^2 = 64 + 25 - 80(0.5) = 49$$ $$c = 7$$

Bearings:

Three-figure bearings measure angles clockwise from North:

• North = 000°
• East = 090°
• South = 180°
• West = 270°

Example: “Bearing of A from B is 045°” means: from B, look 45° clockwise from North to see A.

Projection:

The projection of vector $\vec{a}$ on $\vec{b}$: $$\text{proj}_{\vec{b}} \vec{a} = |\vec{a}|\cos\theta \times \hat{b}$$

Or using components: $$\text{proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|^2}\vec{b}$$

⚡ NABTEB Quick Reference:

• $\sin\theta = \text{opp}/\text{hyp}$; $\cos\theta = \text{adj}/\text{hyp}$; $\tan\theta = \text{opp}/\text{adj}$
• $\sin^2\theta + \cos^2\theta = 1$
• Sine rule: $a/\sin A = b/\sin B = c/\sin C$
• Cosine rule: $a^2 = b^2 + c^2 - 2bc\cos A$
• Area: $A = \frac{1}{2}ab\sin C$
• $180° = \pi$ radians
• Period of $\sin x$ and $\cos x$: 360°; Period of $\tan x$: 180°
• $\sin(A+B) = \sin A\cos B + \cos A\sin B$
• $\cos(A+B) = \cos A\cos B - \sin A\sin B$