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The essential right-angled triangle toolkit for IGCSE Maths. Pythagoras theorem, trigonometric ratios, exact values, and worked examples for grades 5-7.
Pythagoras theorem (a squared plus b squared equals c squared) lets you find missing sides in right-angled triangles. Basic trigonometry (SOH CAH TOA) extends this by using the ratios sin, cos, and tan to find unknown sides and angles. Together they form the foundation for all trigonometry at IGCSE level and appear on both Paper 1 and Paper 2, typically carrying 8-12 marks across a full exam sitting.
Source: Edexcel IGCSE Mathematics (9-1) Specification 4MA1
Pythagoras theorem applies to every right-angled triangle. It states that the square of the hypotenuse (the side opposite the right angle, always the longest side) equals the sum of the squares of the other two sides.
Pythagoras Theorem
a squared + b squared = c squared (where c is the hypotenuse)
If you know the two shorter sides a and b, find the hypotenuse by calculating c = square root of (a squared + b squared). For example, if a = 3 and b = 4, then c = square root of (9 + 16) = square root of 25 = 5.
If you know the hypotenuse c and one shorter side b, rearrange to a squared = c squared - b squared, then take the square root. For example, if c = 13 and b = 5, then a = square root of (169 - 25) = square root of 144 = 12.
SOH CAH TOA gives you three ratios that link an angle in a right-angled triangle to pairs of sides:
| Mnemonic | Ratio | Formula |
|---|---|---|
| SOH | Sine | sin(angle) = Opposite / Hypotenuse |
| CAH | Cosine | cos(angle) = Adjacent / Hypotenuse |
| TOA | Tangent | tan(angle) = Opposite / Adjacent |
To find a missing side, follow these steps:
When you know two sides but need to find an angle, use the inverse trigonometric functions on your calculator: sin inverse, cos inverse, or tan inverse (sometimes written as arcsin, arccos, arctan).
The process is the same as finding a side, except at the final step you apply the inverse function. For example, if you know the opposite side is 7 and the hypotenuse is 10:
Always check your calculator is in degree mode, not radians. If your answer seems unreasonable (e.g. an angle greater than 90 degrees in a right-angled triangle), check your mode setting first.
The IGCSE specification requires you to know the exact values of sin, cos, and tan for the key angles. These are tested on Paper 1 (non-calculator), so you cannot rely on your calculator to find them.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0 degrees | 0 | 1 | 0 |
| 30 degrees | 1/2 | root 3 / 2 | 1 / root 3 |
| 45 degrees | root 2 / 2 | root 2 / 2 | 1 |
| 60 degrees | root 3 / 2 | 1/2 | root 3 |
| 90 degrees | 1 | 0 | undefined |
Notice the pattern: the sin values for 0, 30, 45, 60, 90 degrees follow the sequence 0, 1/2, root 2 / 2, root 3 / 2, 1. The cos values are the same sequence in reverse. Learning this symmetry makes the table much easier to recall under exam conditions.
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the length of the hypotenuse.
Step 1 (Write Pythagoras formula):
c squared = a squared + b squared
Step 2 (Substitute):
c squared = 6 squared + 8 squared = 36 + 64 = 100
Step 3 (Square root):
c = square root of 100 = 10 cm
Answer: 10 cm
In a right-angled triangle, the side opposite angle x is 9 cm and the hypotenuse is 15 cm. Find angle x. Give your answer correct to 1 decimal place.
Step 1 (Label sides):
Opposite = 9 cm, Hypotenuse = 15 cm
Step 2 (Choose ratio):
We have Opposite and Hypotenuse, so use SOH: sin(x) = Opposite / Hypotenuse
Step 3 (Substitute):
sin(x) = 9 / 15 = 0.6
Step 4 (Inverse sin):
x = sin inverse (0.6) = 36.8698...
Answer: x = 36.9 degrees (1 d.p.)
A right-angled triangle has a hypotenuse of 8 cm and one angle of 60 degrees. Find the exact length of the side adjacent to the 60-degree angle. Do not use a calculator.
Step 1 (Label sides):
Adjacent = unknown, Hypotenuse = 8 cm, angle = 60 degrees
Step 2 (Choose ratio):
We have Adjacent and Hypotenuse, so use CAH: cos(60) = Adjacent / 8
Step 3 (Use exact value):
cos(60) = 1/2 (from the exact values table)
Step 4 (Solve):
1/2 = Adjacent / 8
Adjacent = 8 x 1/2 = 4 cm
Answer: 4 cm (exact)
Squaring then adding instead of adding then square-rooting: Students sometimes calculate a + b and then square, rather than computing a squared + b squared first, then taking the square root. Follow the order of operations carefully.
Using the wrong trig ratio: Mislabelling sides is the root cause. Always label Opposite, Adjacent, and Hypotenuse relative to the specific angle you are working with before choosing SOH, CAH, or TOA.
Calculator in radian mode: If your inverse trig answer is a very small number (like 0.64 instead of 36.9 degrees), your calculator is almost certainly in radian mode. Switch to degree mode before the exam.
Not knowing exact trig values on Paper 1: On the non-calculator paper, you must recall exact values from memory. If you leave an answer as sin(30) instead of writing 1/2, you will lose the mark. Memorise the table before exam day.
Decide Pythagoras or trig first: If the question gives two sides and no angle, use Pythagoras. If it gives an angle and one side (or asks for an angle), use trigonometry. Making this decision first saves time and prevents choosing the wrong method.
Label all three sides before choosing a ratio: Write O, A, and H on the diagram relative to the angle. This makes it impossible to pick the wrong ratio and earns credit for clear working.
Sense-check your answer: The hypotenuse must always be the longest side. If your calculated side is longer than the hypotenuse, you have made an error. Similarly, angles in a right-angled triangle must be between 0 and 90 degrees (for the non-right angles).
Learn exact values by pattern: The sin values from 0 to 90 degrees go 0, 1/2, root 2 / 2, root 3 / 2, 1. Cos is the same list in reverse. Tan is sin divided by cos. Knowing this pattern means you only need to memorise one row of the table.
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