Right-angled triangles

1. Overview

Right-angled triangles are the foundation for trigonometry and are essential for solving problems involving angles and lengths. This topic covers Pythagoras’ Theorem and the trigonometric ratios (sine, cosine, and tangent) which allow you to calculate unknown sides and angles in right-angled triangles. These skills are crucial for various applications, including navigation, surveying, and construction. Mastering these concepts will enable you to tackle 2D problems and prepare you for more complex 3D applications.

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Key Definitions

• Hypotenuse: The longest side of a right-angled triangle, always opposite the 90° angle.
• Opposite: The side directly across from a specific marked angle ($\theta$).
• Adjacent: The side next to the marked angle ($\theta$) that is not the hypotenuse.
• Pythagoras’ Theorem: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
• SOH CAH TOA: A mnemonic used to remember the three trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

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Core Content

A. Pythagoras’ Theorem

Used to find the length of an unknown side in a right-angled triangle when you know the lengths of the other two sides.

Pythagoras' Theorem (given on formula sheet): $\qquad a^2 + b^2 = c^2$

Worked example 1 — Finding the Hypotenuse

Question: A right-angled triangle has sides of length 6 cm and 8 cm. Calculate the length of the hypotenuse.

1. State the formula: $a^2 + b^2 = c^2$
2. Substitute the values: $6^2 + 8^2 = c^2$
3. Calculate squares: $36 + 64 = c^2$
4. Add: $100 = c^2$
5. Square root both sides: $\sqrt{100} = c$
6. Final Answer: $c = \textbf{10 cm}$

Worked example 2 — Finding a Shorter Side

Question: The hypotenuse of a right-angled triangle is 13 m, and one of the shorter sides is 5 m. Find the length of the other side.

1. State the formula: $a^2 + b^2 = c^2$
2. Substitute the values: $5^2 + b^2 = 13^2$
3. Rearrange for $b^2$: $b^2 = 13^2 - 5^2$
4. Calculate squares: $b^2 = 169 - 25$
5. Subtract: $b^2 = 144$
6. Square root both sides: $b = \sqrt{144}$
7. Final Answer: $b = \textbf{12 m}$

Worked example 3 — Real-world Pythagoras

Question: A rectangular gate is 3 meters wide and 4 meters high. A diagonal support bar is added to the gate. How long is the support bar?

1. Recognize the problem: The gate and support bar form a right-angled triangle, where the support bar is the hypotenuse.
2. State the formula: $a^2 + b^2 = c^2$
3. Substitute the values: $3^2 + 4^2 = c^2$
4. Calculate squares: $9 + 16 = c^2$
5. Add: $25 = c^2$
6. Square root both sides: $c = \sqrt{25}$
7. Final Answer: $c = \textbf{5 meters}$

B. Trigonometric Ratios (SOH CAH TOA)

Used when the problem involves an angle and two sides.

• $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ (SOH)
• $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ (CAH)
• $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ (TOA)

Worked example 4 — Finding a Missing Side

Question: In a right-angled triangle, the hypotenuse is 10 cm and one angle is 60°. Calculate the length of the side opposite the 60° angle.

1. Identify the sides: We have the Hypotenuse (10 cm) and want the Opposite side ($x$).
2. Choose the ratio: SOH (Sine = Opposite/Hypotenuse) relates these sides.
3. Set up the equation: $\sin(60°) = \frac{x}{10}$
4. Rearrange to solve for $x$: $x = 10 \times \sin(60°)$
5. Calculate: $x = 10 \times 0.866025...$ (keep at least 4 significant figures)
6. Final Answer: $x = \textbf{8.66 cm}$ (to 3 significant figures).

Worked example 5 — Finding a Missing Angle

Question: A ramp is 8 m long and rises to a height of 1 m. Find the angle of elevation, $\theta$, of the ramp.

1. Identify the sides: The ramp is the Hypotenuse (8 m). The height is the Opposite (1 m) to the angle $\theta$.
2. Choose the ratio: SOH (Sine = Opposite/Hypotenuse) relates these sides.
3. Set up the equation: $\sin(\theta) = \frac{1}{8}$
4. Use the inverse function ($\sin^{-1}$) to find the angle: $\theta = \sin^{-1}(\frac{1}{8})$
5. Calculate: $\theta = \sin^{-1}(0.125) = 7.1807...$
6. Final Answer: $\theta = \textbf{7.2°}$ (to 1 decimal place).

Worked example 6 — Combining Pythagoras and Trig

Question: In a right-angled triangle ABC, angle A = 30°, and the side adjacent to angle A (AB) is 12 cm. Find the length of the hypotenuse (AC) and the side opposite angle A (BC).

1. Finding the Hypotenuse (AC):

   • Identify the sides: We have the Adjacent (12 cm) and want the Hypotenuse (AC).
   • Choose the ratio: CAH (Cosine = Adjacent/Hypotenuse) relates these sides.
   • Set up the equation: $\cos(30°) = \frac{12}{AC}$
   • Rearrange to solve for AC: $AC = \frac{12}{\cos(30°)}$
   • Calculate: $AC = \frac{12}{0.866025...} = 13.8564...$
   • Final Answer: $AC = \textbf{13.9 cm}$ (to 3 significant figures).

2. Finding the Opposite side (BC):

   • Method 1 (using Tangent):
     • Identify the sides: We have the Adjacent (12 cm) and want the Opposite (BC).
     • Choose the ratio: TOA (Tangent = Opposite/Adjacent) relates these sides.
     • Set up the equation: $\tan(30°) = \frac{BC}{12}$
     • Rearrange to solve for BC: $BC = 12 \times \tan(30°)$
     • Calculate: $BC = 12 \times 0.577350... = 6.9282...$
     • Final Answer: $BC = \textbf{6.93 cm}$ (to 3 significant figures).
   • Method 2 (using Pythagoras):
     • We know AC = 13.8564... cm and AB = 12 cm.
     • State the formula: $AB^2 + BC^2 = AC^2$
     • Substitute the values: $12^2 + BC^2 = 13.8564...^2$
     • Rearrange for $BC^2$: $BC^2 = 13.8564...^2 - 12^2$
     • Calculate squares: $BC^2 = 192.000... - 144$
     • Subtract: $BC^2 = 48$
     • Square root both sides: $BC = \sqrt{48}$
     • Final Answer: $BC = \textbf{6.93 cm}$ (to 3 significant figures).

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Extended Content (Extended Only)

While topic 6.2 is primarily Core, Extended students must apply these trigonometric principles in more complex scenarios, particularly in three-dimensional problems. This often involves visualizing and extracting right-angled triangles from 3D shapes like cuboids, pyramids, and prisms. A common example is finding the angle between a line and a plane, or the angle between two planes. These problems require careful spatial reasoning and the ability to break down the 3D geometry into simpler 2D right-angled triangles.

For example, consider a cuboid with dimensions 5 cm x 4 cm x 3 cm. To find the angle between the space diagonal (connecting opposite corners of the cuboid) and the base, you would first need to calculate the length of the diagonal of the base using Pythagoras' theorem. Then, you would use trigonometry (specifically the tangent ratio) in a right-angled triangle formed by the height of the cuboid, the base diagonal, and the space diagonal to find the required angle. Remember to maintain accuracy throughout your calculations by keeping several decimal places until the final rounding step.

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Key Equations

Formula	Meaning	Unit Notes
$\mathbf{a^2 + b^2 = c^2}$	Pythagoras' Theorem ($c$ is hypotenuse)	Lengths must be in the same units.
$\mathbf{\sin(\theta) = \frac{O}{H}}$	Sine ratio	Calculator must be in DEGREES mode.
$\mathbf{\cos(\theta) = \frac{A}{H}}$	Cosine ratio	Calculator must be in DEGREES mode.
$\mathbf{\tan(\theta) = \frac{O}{A}}$	Tangent ratio	Calculator must be in DEGREES mode.

Note: All formulas in this table are provided on the IGCSE Mathematics formula sheet.

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Common Mistakes to Avoid

• ❌ Incorrect Pythagoras Application: Forgetting to subtract when finding a shorter side.
  • ✓ Right: If finding a shorter side, rearrange Pythagoras' Theorem to $a^2 = c^2 - b^2$, where $c$ is the hypotenuse.
• ❌ Premature Rounding: Rounding intermediate values (e.g., $\sin(23°)$) to fewer than 4 significant figures.
  • ✓ Right: Store intermediate values in your calculator's memory or use the "ANS" button to maintain accuracy until the final step. Only round the final answer to the required degree of accuracy.
• ❌ Assuming a Right Angle: Applying SOH CAH TOA or Pythagoras' Theorem to triangles that are not explicitly stated to be right-angled.
  • ✓ Right: Verify that the triangle has a marked right angle or that the problem provides information that allows you to deduce the presence of a right angle before using these methods.
• ❌ Incorrect Calculator Mode: Having your calculator in radians (RAD) or gradians (GRAD) instead of degrees (DEG).
  • ✓ Right: Always double-check that your calculator is in degree mode (DEG or D) before starting any trigonometric calculations.
• ❌ Confusing Opposite and Adjacent: Misidentifying the opposite and adjacent sides relative to the given angle.
  • ✓ Right: Carefully highlight the angle in question and then clearly label the opposite and adjacent sides based on their position relative to that angle.

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Exam Tips

• Calculator Mode: Before starting any trigonometry question, ensure your calculator is in degree mode ("D" or "DEG" should be displayed).
• Show Your Working: Always show your steps, even if you use your calculator extensively. This allows you to earn method marks even if you make a calculation error.
• Accuracy: Unless the question specifies otherwise, give angles to 1 decimal place and side lengths to 3 significant figures.
• Command Words: Pay close attention to command words:
  • "Calculate/Find": Requires full working and a numerical answer.
  • "Show that": You are given the answer; you must demonstrate every step of your calculation to arrive at the given answer.
• Real-World Contexts: Be prepared for questions involving angles of elevation and depression. Always draw a clear diagram to represent the situation. The angle of elevation is measured upwards from the horizontal, while the angle of depression is measured downwards from the horizontal.
• Significant Figures: Remember to round your final answer to the specified number of significant figures. If no specific requirement is given, use 3 significant figures for lengths.
• Units: Always include the correct units in your final answer (e.g., cm, m, °).