“Trigonometry: Sine, Cosine and Tangent”

🟢 Lite — Quick Review (1h–1d)

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

Right-Angled Triangle Definitions:

For angle $\theta$ in a right-angled triangle: $$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$$ $$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$$ $$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{\sin\theta}{\cos\theta}$$

SOHCAHTOA:

• Sine = Opposite ÷ Hypotenuse
• Cosine = Adjacent ÷ Hypotenuse
• Tangent = Opposite ÷ Adjacent

Remember: $\tan\theta = \frac{\sin\theta}{\cos\theta}$, so $\sin$ and $\cos$ are NEVER zero in the denominator.

Special Angles:

Angle	sin	cos	tan
0°	0	1	0
30°	1/2	√3/2	1/√3 = √3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

⚡ WAEC Tip: For 30° and 60°, remember: $\sin 30 = \cos 60 = \frac{1}{2}$, $\cos 30 = \sin 60 = \frac{\sqrt{3}}{2}$. Draw an equilateral triangle split in half to derive these values.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Solving Right-Angled Triangle Problems:

Worked Example: A ladder 5m long leans against a wall making 60° with the ground. Find:

1. Height of the wall
2. Distance of ladder foot from wall

Solution:

1. $\sin 60° = \frac{h}{5}$ $h = 5 \times \sin 60° = 5 \times \frac{\sqrt{3}}{2} = \frac{5\sqrt{3}}{2} ≈ 4.33,\text{m}$

2. $\cos 60° = \frac{d}{5}$ $d = 5 \times \cos 60° = 5 \times \frac{1}{2} = 2.5,\text{m}$

Angles of Elevation and Depression:

• Angle of elevation: Angle measured UP from horizontal
• Angle of depression: Angle measured DOWN from horizontal (equal to angle of elevation if looking at same point from opposite direction)

Worked Example: A man 1.8m tall stands 10m from a tree. The angle of elevation to the top is 30°. Find tree height.

$\tan 30° = \frac{h}{10}$ where $h$ is height above man’s eyes $h = 10 \times \frac{1}{\sqrt{3}} = \frac{10}{\sqrt{3}} ≈ 5.77,\text{m}$ Tree height = $5.77 + 1.8 = 7.57,\text{m}$

The Trigonometric Ratios in All Quadrants:

Quadrant	Angles	sin	cos	tan
I	0° to 90°	+	+	+
II	90° to 180°	+	-	-
III	180° to 270°	-	-	+
IV	270° to 360°	-	+	-

Reference Angles: The acute angle made with the nearest x-axis.

• Reference angle for 150° = 30° (since 180° - 150° = 30°)
• $\sin 150° = \sin 30° = +\frac{1}{2}$ (QII, sin positive)
• $\cos 150° = -\cos 30° = -\frac{\sqrt{3}}{2}$ (QII, cos negative)

⚡ Common Mistake: Forgetting that $\sin 30° = \frac{1}{2}$ but $\sin 150°$ is NOT the same. Always check the quadrant for the correct sign.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

The Sine Rule:

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

Use when you know:

• Two angles and one side (AAS or ASA)
• Two sides and an angle opposite one of them (SSA)

Worked Example: Find side $a$ if $A = 30°$, $B = 45°$, and $b = 10,\text{cm}$.

$\frac{a}{\sin 30°} = \frac{10}{\sin 45°}$ $a = 10 \times \frac{\sin 30°}{\sin 45°} = 10 \times \frac{0.5}{0.707} ≈ 7.07,\text{cm}$

The Cosine Rule:

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

Use when you know:

• Two sides and the included angle (SAS)
• All three sides (SSS) — find angles

Worked Example: Find angle $A$ if $a = 7$, $b = 5$, $c = 6$.

$\cos A = \frac{5^2 + 6^2 - 7^2}{2 \times 5 \times 6} = \frac{25 + 36 - 49}{60} = \frac{12}{60} = 0.2$ $A = \cos^{-1}(0.2) ≈ 78.5°$

Area of a Triangle:

$$K = \frac{1}{2}ab \sin C$$

Or using $K = \frac{1}{2}bc \sin A = \frac{1}{2}ca \sin B$

3D Trigonometry:

For angles between lines and planes in 3D:

1. Find the angle the line makes with the horizontal (ground)
2. Project the line onto the plane
3. Use right-angled triangle trigonometry

Worked Example: A pole is on a slope inclined at 15° to the horizontal. A wire is attached from a point 20m up the slope to the top of the pole (vertical). The wire makes 50° with the slope. Find height of pole.

In triangle: wire = 20m, angle at base = 50° $h = 20 \times \sin 50° ≈ 15.3,\text{m}$

Trigonometric Identities:

1. $\sin^2\theta + \cos^2\theta = 1$
2. $1 + \tan^2\theta = \sec^2\theta$
3. $1 + \cot^2\theta = \csc^2\theta$
4. $\sin 2\theta = 2\sin\theta\cos\theta$
5. $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
6. $\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$

Solving Trigonometric Equations:

Example: Solve $\sin x = \frac{1}{2}$ for $0° \leq x \leq 360°$

$\sin x = 0.5$ gives $x = 30°$ or $x = 180° - 30° = 150°$

Example: Solve $2\cos x - 1 = 0$ for $0° \leq x \leq 360°$

$2\cos x = 1$, so $\cos x = 0.5$ $x = 60°$ or $x = 360° - 60° = 300°$

⚡ WAEC Previous Year Pattern:

Year	Question	Concept
2023	Find height using angle of elevation	SOHCAHTOA
2022	Solve trig equation	sin x = value
2021	Sine rule application	Non-right triangle

Radians:

$$\pi \text{ radians} = 180°$$ $$1 \text{ radian} = \frac{180°}{\pi} ≈ 57.3°$$

Arc length $s = r\theta$ (where $\theta$ in radians) Area of sector $A = \frac{1}{2}r^2\theta$

Double Angle Formulae Derivation: $\sin(A+B) = \sin A \cos B + \cos A \sin B$

For $\sin 2A$: Let $A = B$ $\sin 2A = \sin A \cos A + \cos A \sin A = 2\sin A \cos A$

⚡ Exam Strategy: In problems with angles of elevation/depression, draw a clear diagram first. Label the ground distance, height, and angle. Use $\sin$ for opposite/hypotenuse, $\cos$ for adjacent/hypotenuse, $\tan$ for opposite/adjacent. Check if your answer is sensible.

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