Pythagoras' Theorem and Trigonometry: GCSE Study Notes
Welcome to your complete guide to Pythagoras' Theorem and Trigonometry! These two mathematical tools are among the most useful in the entire Geometry and Measures curriculum. Whether you are calculating the height of a building, navigating a ship, or designing 3D video game graphics, these formulas are at work behind the scenes.
Don't worry if this topic feels a bit intimidating at first! We will break everything down into clear, easy-to-follow steps with helpful memory tricks and real-life examples.
---Part 1: Pythagoras' Theorem
What is Pythagoras' Theorem?
Pythagoras' theorem is a rule that connects the lengths of all three sides in any right-angled triangle. It states that the square of the longest side is equal to the sum of the squares of the two shorter sides.
Formula:
\(a^2 + b^2 = c^2\)
Here, \(c\) is always the hypotenuse (the longest side, located directly opposite the \(90^\circ\) right angle), while \(a\) and \(b\) are the two shorter sides.
Step-by-Step: How to Label Your Triangle
• Step 1: Find the right angle symbol (\(\llcorner\)).
• Step 2: Draw an arrow directly across from the right angle. The side it points to is the hypotenuse (\(c\)).
• Step 3: Label the other two sides \(a\) and \(b\) (it does not matter which is which).
Case 1: Finding the Hypotenuse (The Longest Side)
When you know the two shorter sides and need to find the longest side, you add the squares:
Method: Square both sides \(\rightarrow\) Add them together \(\rightarrow\) Take the square root.
Example: A triangle has shorter sides of length \(6\text{ cm}\) and \(8\text{ cm}\). Find the hypotenuse \(c\).
\(c^2 = a^2 + b^2\)
\(c^2 = 6^2 + 8^2\)
\(c^2 = 36 + 64 = 100\)
\(c = \sqrt{100} = 10\text{ cm}\)
Case 2: Finding a Shorter Side
When you already know the hypotenuse and need to find one of the shorter sides, you must subtract:
Method: Square the hypotenuse \(\rightarrow\) Square the known shorter side \(\rightarrow\) Subtract \(\rightarrow\) Take the square root.
Rearranged Formula:
\(a^2 = c^2 - b^2\) or \(a = \sqrt{c^2 - b^2}\)
Example: A ladder of length \(13\text{ m}\) rests against a wall. The base of the ladder is \(5\text{ m}\) away from the wall. How high up the wall does the ladder reach?
• Here, the ladder is the hypotenuse (\(c = 13\text{ m}\)) and the base is \(b = 5\text{ m}\).
\(a^2 = 13^2 - 5^2\)
\(a^2 = 169 - 25 = 144\)
\(a = \sqrt{144} = 12\text{ m}\)
Common Mistakes to Avoid with Pythagoras
• Forgetting to square root at the end: An answer of \(144\text{ m}\) for a ladder is impossible if the ladder is only \(13\text{ m}\) long! Always check if your answer looks sensible.
• Adding when you should subtract: Only add when calculating the hypotenuse. If you are calculating a shorter side, always subtract from the square of the hypotenuse.
Key Takeaway for Pythagoras: Finding the long side? Square, Square, Add, Square Root. Finding a short side? Square, Square, Subtract, Square Root.
---Part 2: Right-Angled Trigonometry (SOH CAH TOA)
While Pythagoras uses only side lengths, Trigonometry connects side lengths with angles in right-angled triangles.
Labeling the Triangle Relative to an Angle (\(\theta\))
Before doing any calculation, you must label the three sides correctly based on the position of the given angle \(\theta\) (theta):
• Hypotenuse (\(H\)): The longest side, always opposite the \(90^\circ\) angle.
• Opposite (\(O\)): The side directly across from the angle \(\theta\).
• Adjacent (\(A\)): The side next to the angle \(\theta\), sandwiched between \(\theta\) and the right angle.
The Three Trigonometric Ratios
\(\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H}\)
\(\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H}\)
\(\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A}\)
Memory Trick: Remember the phrase SOH CAH TOA:
• SOH: Sin = Opposite / Hypotenuse
• CAH: Cos = Adjacent / Hypotenuse
• TOA: Tan = Opposite / Adjacent
Calculator Check: Make sure your calculator is in Degrees mode (look for a small 'D' or 'DEG' at the top of the screen). If it says 'R' or 'G', your answers will be incorrect!
Finding a Missing Side
• Step 1: Label the sides (\(O\), \(A\), \(H\)).
• Step 2: Identify the side you know and the side you want to find.
• Step 3: Choose the ratio from SOH CAH TOA that contains both sides.
• Step 4: Substitute values and solve for the unknown side.
Example: A triangle has an angle of \(35^\circ\) and a hypotenuse of \(12\text{ cm}\). Calculate the length of the opposite side \(x\).
• Known: Angle \(= 35^\circ\), Hypotenuse \(H = 12\text{ cm}\)
• Target: Opposite \(O = x\)
• Formula: \(\sin(\theta) = \frac{O}{H}\)
\(\sin(35^\circ) = \frac{x}{12}\)
\(x = 12 \times \sin(35^\circ)\)
\(x \approx 12 \times 0.5736 = 6.88\text{ cm}\) (to 2 decimal places)
Finding a Missing Angle
To find an unknown angle, use the inverse trigonometric functions on your calculator: \(\sin^{-1}\), \(\cos^{-1}\), or \(\tan^{-1}\) (usually found by pressing SHIFT or 2nd before the trig button).
Example: In a right-angled triangle, the adjacent side is \(7\text{ cm}\) and the hypotenuse is \(11\text{ cm}\). Find angle \(\theta\).
• Known: Adjacent \(A = 7\text{ cm}\), Hypotenuse \(H = 11\text{ cm}\)
• Formula: \(\cos(\theta) = \frac{A}{H}\)
\(\cos(\theta) = \frac{7}{11}\)
\(\theta = \cos^{-1}\left(\frac{7}{11}\right)\)
\(\theta \approx 50.5^\circ\) (to 1 decimal place)
Key Takeaway for SOH CAH TOA: Label your sides first, pick the matching formula, and use inverse trig (\(\sin^{-1}, \cos^{-1}, \tan^{-1}\)) when solving for an angle.
---Part 3: Angles of Elevation and Depression
Understanding the Terms
• Angle of Elevation: The angle measured upwards from the horizontal line of sight to an object.
• Angle of Depression: The angle measured downwards from the horizontal line of sight to an object.
Top Tip: The angle of depression is always measured from an imaginary horizontal line, never from the vertical wall or tower! Because horizontal lines are parallel, the angle of depression from the top of a cliff to a boat equals the angle of elevation from the boat to the top of the cliff (alternate 'Z' angles).
---Part 4: Exact Trigonometric Values (Non-Calculator)
For non-calculator exams, you need to know the exact values for certain special angles (\(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\)):
Values for Sine (\(\sin\)):
• \(\sin(0^\circ) = 0\)
• \(\sin(30^\circ) = \frac{1}{2}\)
• \(\sin(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
• \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\)
• \(\sin(90^\circ) = 1\)
Values for Cosine (\(\cos\)):
• \(\cos(0^\circ) = 1\)
• \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
• \(\cos(45^\circ) = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\)
• \(\cos(60^\circ) = \frac{1}{2}\)
• \(\cos(90^\circ) = 0\)
Values for Tangent (\(\tan\)):
• \(\tan(0^\circ) = 0\)
• \(\tan(30^\circ) = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}\)
• \(\tan(45^\circ) = 1\)
• \(\tan(60^\circ) = \sqrt{3}\)
• \(\tan(90^\circ) =\) undefined
Memory Trick: Notice how the sine values go up from \(\frac{\sqrt{0}}{2}\) to \(\frac{\sqrt{4}}{2}\), while the cosine values are the exact same list in reverse order!
---Part 5: 3D Pythagoras and Trigonometry
Pythagoras in 3D
To find the long internal diagonal \(d\) of a cuboid with length \(l\), width \(w\), and height \(h\), you can extend Pythagoras' theorem into three dimensions:
3D Formula:
\(d^2 = l^2 + w^2 + h^2\) \(\implies\) \(d = \sqrt{l^2 + w^2 + h^2}\)
Example: Find the length of the longest stick that can fit inside a box of dimensions \(4\text{ cm} \times 5\text{ cm} \times 12\text{ cm}\).
\(d = \sqrt{4^2 + 5^2 + 12^2} = \sqrt{16 + 25 + 144} = \sqrt{185} \approx 13.6\text{ cm}\)
3D Angles
To find the angle between a 3D line (like an internal diagonal) and a plane (like the floor of a cuboid):
• Step 1: Drop a vertical line from the top point down to the base to create a right-angled triangle.
• Step 2: Calculate the diagonal across the floor base using standard 2D Pythagoras.
• Step 3: Use SOH CAH TOA on the vertical right-angled triangle inside the 3D shape.
Part 6: Non-Right-Angled Triangles (Higher Tier)
When working with triangles that do not contain a right angle, standard SOH CAH TOA does not work. Instead, we use the Sine Rule, the Cosine Rule, and the Trigonometric Area Formula.
Standard Labeling Convention: Capital letters \(A, B, C\) represent the angles, and matching lowercase letters \(a, b, c\) represent the sides opposite those angles.
1. Area of Any Triangle
When you know two sides and the included angle (the angle trapped between them):
\(\text{Area} = \frac{1}{2}ab\sin(C)\)
Example: A triangle has sides of \(7\text{ cm}\) and \(10\text{ cm}\) with an angle of \(40^\circ\) between them.
\(\text{Area} = \frac{1}{2} \times 7 \times 10 \times \sin(40^\circ) = 35 \times 0.6428 \approx 22.5\text{ cm}^2\)
2. The Sine Rule
Use the Sine Rule when you have opposite pairs of sides and angles.
• Finding a side:
\(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\)
• Finding an angle:
\(\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}\)
3. The Cosine Rule
Use the Cosine Rule when you have:
• Three sides and want to find an angle.
• Two sides and the included angle and want to find the third side.
• Finding a side:
\(a^2 = b^2 + c^2 - 2bc\cos(A)\)
• Finding an angle (rearranged):
\(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\)
Quick Decision Guide: Which Rule Should I Use?
• Right-angled triangle?
\(\rightarrow\) Only side lengths involved: Use Pythagoras' Theorem (\(a^2 + b^2 = c^2\)).
\(\rightarrow\) Sides and angles involved: Use SOH CAH TOA.
• Non-right-angled triangle?
\(\rightarrow\) Involves an opposite pair of angle and side: Use the Sine Rule.
\(\rightarrow\) Involves two sides with the trapped angle, or all three sides: Use the Cosine Rule.
\(\rightarrow\) Finding the space inside with two sides and included angle: Use \(\frac{1}{2}ab\sin(C)\).