Trigonometry

Subject: Additional Mathematics
Topic: 7
Cambridge Code: 4037 / 0606

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Right-Angled Triangle Ratios

Trigonometric Ratios - Relationships between sides and angles

The Three Ratios

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{o}{h}$

$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{h}$

$\tan\theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{o}{a}$

Memory Aid: SOH-CAH-TOA

Sine = Opposite/Hypotenuse Cosine = Adjacent/Hypotenuse Tangent = Opposite/Adjacent

Reciprocal Ratios

$\cosec\theta = \frac{1}{\sin\theta} = \frac{h}{o}$

$\sec\theta = \frac{1}{\cos\theta} = \frac{h}{a}$

$\cot\theta = \frac{1}{\tan\theta} = \frac{a}{o}$

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Trigonometric Graphs

Sine Graph: $y = \sin x$

• Period: 360° or $2\pi$ radians
• Amplitude: 1 (range: -1 to 1)
• Zeros: 0°, 180°, 360°, ...
• Maxima: 90°, 450°, ...
• Minima: 270°, 630°, ...

Cosine Graph: $y = \cos x$

• Period: 360° or $2\pi$ radians
• Amplitude: 1
• Zeros: 90°, 270°, ...
• Maxima: 0°, 360°, ...
• Minima: 180°, 540°, ...

Tangent Graph: $y = \tan x$

• Period: 180° or $\pi$ radians
• Range: All real numbers
• Asymptotes: 90°, 270°, ...
• Zeros: 0°, 180°, 360°, ...

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Trigonometric Identities

Pythagorean Identity

$\sin^2\theta + \cos^2\theta = 1$

Quotient Identity

$\tan\theta = \frac{\sin\theta}{\cos\theta}$

Sum and Difference Formulas

$\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$

$\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$

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

Double Angle Formulas

$\sin 2A = 2\sin A \cos A$

$\cos 2A = \cos^2 A - \sin^2 A = 2\cos^2 A - 1 = 1 - 2\sin^2 A$

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

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Sine Rule

Sine Rule - For any triangle:

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

where lowercase letters are sides opposite to angles in capital letters

When to Use

• Two angles and one side known (AAS or ASA)
• Two sides and a non-included angle known (SSA)

Example

In triangle ABC: $A = 30°$, $B = 45°$, $a = 5$ cm Find side $b$:

$\frac{5}{\sin 30°} = \frac{b}{\sin 45°}$

$\frac{5}{0.5} = \frac{b}{\frac{\sqrt{2}}{2}}$

$b = 10 \times \frac{\sqrt{2}}{2} = 5\sqrt{2} \approx 7.07 \text{ cm}$

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Cosine Rule

Cosine Rule - For any triangle:

$a^2 = b^2 + c^2 - 2bc\cos A$

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

When to Use

• All three sides known (SSS)
• Two sides and included angle known (SAS)

Example

In triangle ABC: $a = 7$ cm, $b = 8$ cm, $c = 9$ cm Find angle $A$:

$\cos A = \frac{64 + 81 - 49}{2(8)(9)} = \frac{96}{144} = \frac{2}{3}$

$A = \cos^{-1}\left(\frac{2}{3}\right) \approx 48.2°$

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Area of Triangle

Formula 1: Using base and height $A = \frac{1}{2}bh$

Formula 2: Using two sides and included angle $A = \frac{1}{2}ab\sin C$

Example

Find area of triangle with sides 6 cm, 8 cm and included angle 40°:

$A = \frac{1}{2}(6)(8)\sin 40°$

$= 24 \times 0.643 = 15.4 \text{ cm}^2$

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Solving Trigonometric Equations

General Solutions

For $\sin\theta = k$ where $-1 \leq k \leq 1$:

• Principal angle: $\theta = \sin^{-1}(k)$
• General solutions: $\theta = \sin^{-1}(k) + 360°n$ or $180° - \sin^{-1}(k) + 360°n$

For $\cos\theta = k$:

• Principal angle: $\theta = \cos^{-1}(k)$
• General solutions: $\theta = \pm\cos^{-1}(k) + 360°n$

For $\tan\theta = k$:

• Principal angle: $\theta = \tan^{-1}(k)$
• General solutions: $\theta = \tan^{-1}(k) + 180°n$

Example

Solve $2\sin x = 1$ for $0° \leq x < 360°$

$\sin x = 0.5$ $x = 30° \text{ or } x = 150°$

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Key Points to Remember

1. SOH-CAH-TOA for right triangles
2. Sine rule: use when you have angles/opposite sides
3. Cosine rule: use when sides form a triangle
4. Pythagorean identity: $\sin^2\theta + \cos^2\theta = 1$
5. Double angle formulas very useful
6. Area formula: $A = \frac{1}{2}ab\sin C$

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Worked Examples

Example 1: Sine Rule

Triangle ABC: $a = 10$ m, $\angle A = 40°$, $\angle B = 60°$ Find $b$:

$\frac{10}{\sin 40°} = \frac{b}{\sin 60°}$

$b = \frac{10 \sin 60°}{\sin 40°} = \frac{10 \times 0.866}{0.643} = 13.5 \text{ m}$

Example 2: Cosine Rule

Triangle ABC: $a = 5$ cm, $b = 7$ cm, $C = 85°$ Find $c$:

$c^2 = 25 + 49 - 2(5)(7)\cos 85°$

$c^2 = 74 - 70(0.087) = 68.95$

$c = 8.3 \text{ cm}$

Example 3: Trigonometric Equation

Solve $\tan 2x = 1$ for $0° \leq x < 180°$

$2x = 45° + 180°n$

$x = 22.5°, 112.5°$

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Practice Questions

1. In right triangle: opposite = 5, hypotenuse = 13. Find $\sin\theta$, $\cos\theta$, $\tan\theta$.

2. Solve for $0° \leq x < 360°$:

   • $\sin x = 0.8$
   • $\cos 2x = -0.5$

3. Triangle ABC: $a = 6$ cm, $b = 8$ cm, $c = 10$ cm. Find all angles.

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Revision Tips

• Draw right triangles to determine ratios
• Remember all three sine/cosine/tangent formulas
• Use identities to simplify expressions
• Sine rule for AAS or SSA
• Cosine rule for SAS or SSS
• always consider domain for equations