International Mathematics (0607) | Right-angled triangles

Right-angled Triangles: Your Guide to Trigonometry (IGCSE 0607)

Welcome to the world of Right-angled Triangles! This chapter is your foundation for all of trigonometry. Trigonometry might sound complicated, but it's really just a set of simple, powerful tools that help us figure out unknown lengths and angles in triangles, especially in situations where we can't physically measure them—like finding the height of a skyscraper or the depth of a canyon.

In these notes, we will focus only on triangles that contain a 90° angle. We'll revisit Pythagoras and then unlock the three key trigonometric ratios (Sine, Cosine, and Tangent) and see how they apply in real-world problems.

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1. The Foundation: Pythagoras' Theorem (C7.1)

Before we dive into angles, we must remember our most important rule for calculating sides in a right-angled triangle: Pythagoras' Theorem.

This theorem states that in any right-angled triangle, the square of the length of the longest side (the hypotenuse) is equal to the sum of the squares of the lengths of the other two sides.

The Formula

If \(a\) and \(b\) are the shorter sides (legs) and \(c\) is the hypotenuse:

$$a^2 + b^2 = c^2$$

Key Terminology Review

• Right Angle: The 90° angle, denoted by a small square.
• Hypotenuse (c): The longest side of the triangle. It is *always* the side opposite the right angle.
• Legs (a and b): The two shorter sides that form the right angle.

Key Takeaway: Pythagoras is for calculating lengths only when you know two other lengths.

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2. Introducing the Trigonometric Ratios (C7.2.1)

Pythagoras only works if you know two sides. What if you only know one side and one angle (other than the 90° angle)? That’s where Sine, Cosine, and Tangent come in!

These are relationships (ratios) between the sides and the angles in a right-angled triangle.

2.1 Labelling the Sides

The first and most important step in any trigonometry problem is correctly labeling the sides *relative* to the angle you are using (\(\theta\)).

1. Hypotenuse (H): Always opposite the right angle. (Stays the same!)
2. Opposite (O): The side directly across from the angle \(\theta\).
3. Adjacent (A): The side next to (adjacent to) the angle \(\theta\). (It touches \(\theta\), but isn't the Hypotenuse).

2.2 The Three Ratios: SOH CAH TOA

To remember which ratio uses which sides, we use the famous mnemonic: SOH CAH TOA.

1. Sine (SOH)

SOH stands for: Sine = Opposite / Hypotenuse

$$\sin(\theta) = \frac{O}{H}$$

2. Cosine (CAH)

CAH stands for: Cosine = Adjacent / Hypotenuse

$$\cos(\theta) = \frac{A}{H}$$

3. Tangent (TOA)

TOA stands for: Tangent = Opposite / Adjacent

$$\tan(\theta) = \frac{O}{A}$$

Key Takeaway: SOH CAH TOA tells you which two sides you need to use based on the known angle.

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3. Using Trigonometry to Find an Unknown Side

If you know an acute angle (\(\theta\)) and the length of one side, you can find the length of any other side.

Step-by-Step Guide

1. Label: Mark the known angle \(\theta\). Label the three sides (O, A, H) relative to \(\theta\).
2. Select: Look at which side you know and which side you want to find. Use SOH CAH TOA to choose the ratio that involves these two sides.
3. Substitute: Write the chosen formula and plug in the known values.
4. Solve: Rearrange the equation to find the unknown side.

Example: Finding the Opposite Side

A ladder leans against a wall, making a 70° angle with the ground. If the ladder is 5 meters long (Hypotenuse), how high up the wall does it reach (Opposite)?

1. Label: \(\theta = 70^\circ\). Hypotenuse (H) = 5 m. Opposite (O) = \(x\) (unknown).

2. Select: We have O and H. Use SOH (Sine).

3. Substitute: \(\sin(70^\circ) = \frac{x}{5}\)

4. Solve (rearrange):
$$x = 5 \times \sin(70^\circ)$$

$$x \approx 4.6984...$$

5. Round: \(x = 4.70\) m (3 significant figures, standard accuracy).

Key Takeaway: To find a side, you need one angle and one side. Choose the ratio that uses the two relevant sides.

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4. Using Trigonometry to Find an Unknown Angle

If you know the lengths of two sides, you can find the size of either acute angle.

4.1 Inverse Trigonometric Functions

When you want to find the angle itself, you use the inverse functions, written on your calculator as \(\sin^{-1}\), \(\cos^{-1}\), and \(\tan^{-1}\).

Think of it this way: if \(\sin(\theta) = 0.5\), then \(\theta = \sin^{-1}(0.5)\). The inverse function cancels out the ratio to leave you with the angle.

Step-by-Step Guide

1. Label: Label the three sides (O, A, H) relative to the unknown angle (\(\theta\)) you want to find.
2. Select: Look at the two sides you know. Use SOH CAH TOA to choose the ratio that involves these two known sides.
3. Substitute: Write the chosen formula and plug in the known lengths.
4. Solve: Use the inverse function (\(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\)) on your calculator to find \(\theta\).

Example: Finding the Angle using Tangent

A ramp is 12 m long horizontally (Adjacent) and rises 3 m vertically (Opposite). Find the angle of the ramp, \(\theta\).

1. Label: Opposite (O) = 3 m. Adjacent (A) = 12 m. \(\theta\) is unknown.

2. Select: We have O and A. Use TOA (Tangent).

3. Substitute: \(\tan(\theta) = \frac{3}{12}\)

4. Solve:
$$\theta = \tan^{-1} \left(\frac{3}{12}\right)$$

$$\theta = \tan^{-1} (0.25)$$

$$\theta \approx 14.036...$$

5. Round: \(\theta = 14.0^\circ\) (Angles in degrees should be correct to one decimal place, unless otherwise specified).

Key Takeaway: To find an angle, you need two sides and must use the inverse trig functions (\(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\)).

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5. Real-World Applications (C7.2.2 & E7.2.4)

Trigonometry is essential for solving problems in 2D space, often involving angles related to the ground or horizon.

5.1 Angles of Elevation and Depression (E7.2.4)

These angles help describe looking up or looking down from a horizontal line.

Angle of Elevation

This is the angle measured upwards from the horizontal line of sight to an object above.

Analogy: Think of an Elevator going UP.

Angle of Depression

This is the angle measured downwards from the horizontal line of sight to an object below.

Crucial Point: If a person on a cliff looks down at a boat, the angle of depression from the cliff top is equal to the angle of elevation from the boat (due to the alternate segment theorem, as the horizontal lines are parallel).

5.2 Solving 2D Problems (C7.2.2)

Often, problems combine Pythagoras and trigonometry in a single diagram, or require knowledge of other geometrical concepts (like bearings or isosceles triangles) to set up the right triangle correctly.

Step-by-Step for Complex Problems

1. Draw/Identify: Sketch the scenario and look for the hidden 90° angle(s).
2. Isolate: If the diagram is complex (e.g., two triangles joined together), break it down and focus on just the right-angled triangle needed for the first calculation.
3. Calculate: Use Pythagoras (if two sides are known) or SOH CAH TOA (if one angle and one side, or two sides, are known).

5.3 Bearings and Right Triangles (C7.2.2 Note)

Bearings are used to describe directions (e.g., in navigation or surveying). They are always:

• Measured from the North line.
• Measured Clockwise.
• Given as three figures (e.g., North-East is \(045^\circ\), South is \(180^\circ\)).

When solving trigonometry problems involving bearings, you often need to use your knowledge of parallel lines (North lines are parallel) or angles on a straight line to find the interior angle of the right-angled triangle.

Key Takeaway: Right-angled trigonometry is the mathematical map needed to solve practical 2D problems involving heights, distances, and angles of sight.