Pythagoras’ theorem and trigonometry

Introduction: Your Map to the World of Triangles

Welcome! In this chapter, we are diving into the world of Pythagoras’ Theorem and Trigonometry. These are some of the most powerful tools in Mathematics. Why? Because triangles are everywhere—from the rooftops of houses to the way your GPS calculates your location.

Don't worry if you’ve found this topic intimidating before. We are going to break it down into simple, bite-sized steps. By the end of these notes, you’ll be able to calculate heights of buildings and navigate like a pro!

1. Pythagoras’ Theorem: The "Right-Angle" Legend

Pythagoras’ theorem only works for right-angled triangles (triangles with a \(90^\circ\) angle). Before we use the formula, we must identify the most important side: the Hypotenuse.

Finding the Hypotenuse

The hypotenuse is the longest side of a right-angled triangle. It is always directly opposite the \(90^\circ\) angle. Think of it as the "big boss" of the triangle!

The Formula

In a right-angled triangle with sides \(a\), \(b\), and hypotenuse \(c\):
\(a^2 + b^2 = c^2\)

How to use it:

1. To find the hypotenuse (\(c\)): Square both short sides, add them together, then take the square root.
Example: If \(a = 3\) and \(b = 4\), then \(c^2 = 3^2 + 4^2 = 9 + 16 = 25\). So, \(c = \sqrt{25} = 5\).

2. To find a shorter side (\(a\) or \(b\)): Square the hypotenuse, subtract the square of the other known side, then take the square root.
Example: If \(c = 10\) and \(a = 6\), then \(b^2 = 10^2 - 6^2 = 100 - 36 = 64\). So, \(b = \sqrt{64} = 8\).

Determining if a triangle is Right-Angled

You can use this theorem in reverse! If you are given three sides, check if \(a^2 + b^2\) actually equals \(c^2\). If it does, the triangle is right-angled. If it doesn't, it’s not!

Quick Review:
- Hypotenuse = Longest side (opposite \(90^\circ\)).
- Formula: \(a^2 + b^2 = c^2\).
- Common Mistake: Forgetting to "square root" at the very end. Always check if your answer looks sensible compared to the other sides!

2. Trigonometric Ratios: SOH CAH TOA

When we know an angle and a side, Pythagoras isn't enough. We need Trigonometry. First, we must label the sides relative to the angle (\(\theta\)) we are looking at:

1. Hypotenuse (H): The longest side.
2. Opposite (O): The side directly across from angle \(\theta\).
3. Adjacent (A): The side next to angle \(\theta\) (that isn't the hypotenuse).

The "SOH CAH TOA" Mnemonic

This is your best friend for remembering the ratios:
- SOH: \(\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}\)
- CAH: \(\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}\)
- TOA: \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\)

Finding an Angle

If you need to find the angle itself, you use the "inverse" buttons on your calculator: \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\).
Example: If \(\sin \theta = 0.5\), then \(\theta = \sin^{-1}(0.5) = 30^\circ\).

Did you know? The word "Trigonometry" comes from Greek words meaning "triangle measurement."

3. Extending to Obtuse Angles

In your O-Level syllabus, you need to know what happens to Sine and Cosine when the angle is obtuse (between \(90^\circ\) and \(180^\circ\)).

1. Sine: The sine of an obtuse angle is positive and equal to the sine of its supplementary angle.
\(\sin \theta = \sin(180^\circ - \theta)\)
Example: \(\sin 150^\circ = \sin 30^\circ\).

2. Cosine: The cosine of an obtuse angle is negative.
\(\cos \theta = -\cos(180^\circ - \theta)\)
Example: \(\cos 120^\circ = -\cos 60^\circ\).

Takeaway: If your calculator gives you a negative value for \(\cos \theta\), don't panic! It just means the angle is obtuse.

4. Rules for ANY Triangle (Sine and Cosine Rules)

What if the triangle doesn't have a right angle? We use these two powerful laws. For these formulas, we label sides with lowercase letters (\(a, b, c\)) and their opposite angles with uppercase letters (\(A, B, C\)).

The Sine Rule

Use this when you have "pairs" of sides and opposite angles.
\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)

The Cosine Rule

Use this for "SAS" (Side-Angle-Side) or "SSS" (Side-Side-Side) situations.
To find a side: \(a^2 = b^2 + c^2 - 2bc \cos A\)
To find an angle: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\)

Area of a Triangle

Forget "half base times height" for a moment. If you know two sides and the included angle (the angle between them), use:
Area = \(\frac{1}{2}ab \sin C\)

Key Takeaway: Use the Sine Rule if you see two sides and two angles involved. Use the Cosine Rule if you see three sides and one angle involved.

5. Real-World Applications

Trigonometry isn't just on paper; it's used in 2D and 3D space for navigation and construction.

Angles of Elevation and Depression

- Angle of Elevation: The angle looking up from the horizontal line.
- Angle of Depression: The angle looking down from the horizontal line.
Analogy: Imagine standing on a cliff. Looking at a boat is depression; the boat looking up at you is elevation. Important: These two angles are always equal because they are alternate angles!

Bearings

Bearings are a way to describe direction. Always remember the 3 Golden Rules:
1. Measured from North.
2. Measured Clockwise.
3. Written as 3 digits (e.g., \(045^\circ\) instead of \(45^\circ\)).

3D Problems

When solving 3D problems (like finding the angle a pole makes with the ground):
1. Identify a right-angled triangle inside the 3D shape.
2. Draw that triangle in 2D (flat on your paper).
3. Use Pythagoras or SOH CAH TOA as usual.

Summary Checklist

Before your exam, make sure you can:
- Use \(a^2 + b^2 = c^2\) for right-angled triangles.
- Use SOH CAH TOA for right-angled triangles.
- Use \(\sin \theta = \sin(180^\circ - \theta)\) for obtuse angles.
- Use the Sine Rule and Cosine Rule for non-right-angled triangles.
- Calculate the Area using \(\frac{1}{2}ab \sin C\).
- Draw Bearings from North in a clockwise direction.
- Sketch 2D triangles from 3D diagrams.

Final Tip: Keep your calculator in DEG (Degree) mode. If it’s in RAD or GRAD, your answers will be wrong! Always check the screen for a little "D".