Welcome to Pythagoras and Trigonometry!
Have you ever wondered how architects design slanted roofs, how sailors find their way across the ocean, or even how video games calculate where a character is jumping? They all use the power of triangles! In this chapter, we will master Pythagoras’ theorem and Trigonometry. These tools allow us to find missing lengths and angles in shapes, moving from simple 2D drawings to complex 3D objects.
Note for your exams: You will be provided with a formula sheet (Exam Aid) during your GCSE exams in 2027 and 2028. This sheet includes the basic Pythagoras and Trigonometry formulas, but you still need to know how and when to use them!
1. Pythagoras' Theorem (Foundation and Higher)
Pythagoras' theorem is used only for right-angled triangles. It helps us find a missing side length when we already know the other two sides.
Labeling the Triangle
In a right-angled triangle, the longest side is always opposite the right angle. This side is called the hypotenuse and we label it as \(c\). The two shorter sides are labeled \(a\) and \(b\).
The Formula
\(a^2 + b^2 = c^2\)
This means: "The square of side \(a\) plus the square of side \(b\) equals the square of the hypotenuse \(c\)."
How to solve it: Step-by-Step
To find the longest side (\(c\)):
1. Square both short sides (\(a^2\) and \(b^2\)).
2. Add the results together.
3. Square root the answer: \(c = \sqrt{a^2 + b^2}\).
To find a shorter side (\(a\) or \(b\)):
1. Square the hypotenuse (\(c^2\)) and the known short side.
2. Subtract the smaller square from the larger square.
3. Square root the answer: \(a = \sqrt{c^2 - b^2}\).
Example: A triangle has short sides of \(3cm\) and \(4cm\). To find the hypotenuse: \(3^2 + 4^2 = 9 + 16 = 25\). The \(\sqrt{25} = 5cm\).
Key Takeaway: If you are looking for the longest side, you add. If you are looking for a shorter side, you subtract.
2. Trigonometry: SOH CAH TOA (Foundation and Higher)
Trigonometry (or "Trig") also works with right-angled triangles but involves angles as well as sides.
Labeling the Sides
Before you start, you must label the sides relative to the angle (\(\theta\)) you are using:
1. Hypotenuse (H): The longest side, opposite the right angle.
2. Opposite (O): The side directly across from the angle \(\theta\).
3. Adjacent (A): The side next to the angle \(\theta\) (that isn't the hypotenuse).
The Three Ratios
We use the mnemonic SOH CAH TOA to remember the formulas:
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 a missing angle, you must use the "inverse" buttons on your calculator: \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\).
Example: If \(\sin(\theta) = 0.5\), then \(\theta = \sin^{-1}(0.5) = 30^\circ\).
Common Mistake: Make sure your calculator is in Degree (D) mode, not Radians (R) or Gradians (G)!
3. Exact Trigonometric Values (Foundation and Higher)
For Paper 1 (the non-calculator paper), you are expected to know the exact values for certain angles. You should memorize this table or learn the "hand trick" to derive them.
For \(\sin\):
\(\sin(0^\circ) = 0\)
\(\sin(30^\circ) = \frac{1}{2}\)
\(\sin(45^\circ) = \frac{\sqrt{2}}{2}\)
\(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
\(\sin(90^\circ) = 1\)
For \(\cos\):
\(\cos(0^\circ) = 1\)
\(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
\(\cos(45^\circ) = \frac{\sqrt{2}}{2}\)
\(\cos(60^\circ) = \frac{1}{2}\)
\(\cos(90^\circ) = 0\)
For \(\tan\):
\(\tan(0^\circ) = 0\)
\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)
\(\tan(45^\circ) = 1\)
\(\tan(60^\circ) = \sqrt{3}\)
Did you know? The values for \(\sin\) and \(\cos\) are just mirrors of each other! Notice how \(\sin(30^\circ)\) is the same as \(\cos(60^\circ)\).
4. Pythagoras and Trig in 3D (Higher Tier Only)
(H) In 3D problems, you are usually looking for a length or angle inside a box (cuboid) or a pyramid. The trick is to find a 2D triangle hidden inside the 3D shape.
Steps for 3D Pythagoras:
1. Identify the "floor" diagonal of the shape and use Pythagoras to find its length.
2. Use that "floor" diagonal as one side of a new vertical triangle that goes through the middle of the shape.
3. Use Pythagoras again on this second triangle.
Steps for 3D Trig:
1. Identify the right-angled triangle that contains the angle you need.
2. Often, you will need to use Pythagoras first to find a side length before you can use SOH CAH TOA.
5. Advanced Trigonometry: Non-Right-Angled Triangles (Higher Tier Only)
(H) When a triangle does not have a right angle, we use the Sine Rule or the Cosine Rule. For these, we label the angles with capital letters \(A, B, C\) and the sides opposite them with lowercase letters \(a, b, c\).
The Sine Rule
Used when you have a matching pair (a side and its opposite angle) plus one other piece of information.
To find a side: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
To find an angle: \(\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}\)
The Cosine Rule
Used in two specific cases:
1. SAS: You know two Sides, the Angle between them, and want the third Side.
2. SSS: You know all three Sides and want to find an angle.
Formula: \(a^2 = b^2 + c^2 - 2bc \cos A\)
Area of Any Triangle
If you know two sides and the angle between them (SAS), you can find the area without knowing the height:
Area = \(\frac{1}{2} ab \sin C\)
Quick Review:
- Pythagoras: Only sides, only right-angled.
- SOH CAH TOA: Sides and angles, only right-angled.
- (H) Sine/Cosine Rules: Sides and angles, any triangle.
Common Mistakes to Avoid
1. Mixing up O and A: Always label the Opposite side first (across from the angle) to avoid confusion.
2. Forgetting to Square Root: In Pythagoras, students often find \(c^2\) and forget to take the \(\sqrt{}\) at the end.
3. Rounding too early: Keep the full number in your calculator until the very final step to ensure your answer is accurate.
4. Hypotenuse Confusion: Remember, the hypotenuse is always the longest side. If your calculation for a shorter side results in a number longer than the hypotenuse, you've made a mistake!
Don't worry if this seems tricky at first! Like any tool, Pythagoras and Trigonometry just take a little practice to get used to. Once you can label the triangle correctly, the rest is just following the recipe.