Pythagoras and right-triangle trigonometry

Use Pythagoras' theorem and sine, cosine and tangent to find sides and angles in right-angled triangles.

Also known as: Pythagoras' theorem, SOH CAH TOA, sine cosine tangent.

SAT · Geometry and Trigonometry GCSE Maths · Geometry and measures IGCSE Maths · Trigonometry

Key ideas

Pythagoras: $a^2 + b^2 = c^2$ , where $c$ is the hypotenuse.
$\sin\theta = \frac{\text{opp}}{\text{hyp}}$ , $\cos\theta = \frac{\text{adj}}{\text{hyp}}$ , $\tan\theta = \frac{\text{opp}}{\text{adj}}$ .
Use inverse functions (sin⁻¹, cos⁻¹, tan⁻¹) to find angles.
In a right triangle, the sine of one acute angle equals the cosine of the other.

Pythagoras' theorem

In a right-angled triangle with hypotenuse $c$ (the longest side, opposite the right angle): $a^2 + b^2 = c^2.$ Legs $6$ and $8$ give $c = \sqrt{36 + 64} = 10$ . To find a shorter side, subtract: $\sqrt{13^2 - 5^2} = 12$ .

SOH CAH TOA

Label the sides relative to the angle $\theta$ : opposite, adjacent and hypotenuse. Then

$\sin\theta = \frac{O}{H}, \qquad \cos\theta = \frac{A}{H}, \qquad \tan\theta = \frac{O}{A}.$

With an angle of $30°$ and hypotenuse $14$ , the opposite side is $14\sin 30° = 7$ .

Finding angles and complementary angles

If $\sin\theta = 0.92$ , then $\theta = \sin^{-1}(0.92) \approx 66.9°$ .

The two acute angles of a right triangle add to $90°$ , so $\sin A = \cos B$ when $A + B = 90°$ . Make sure your calculator is in degrees.

Worked examples

Example 1 · Ladder problem

A 5 m ladder reaches 4.6 m up a vertical wall. Find the angle between the ladder and the ground, to the nearest degree.

Step 1.The height is opposite the angle; the ladder is the hypotenuse: $\sin\theta = \frac{4.6}{5} = 0.92$ .
Step 2. $\theta = \sin^{-1}(0.92) \approx 66.9°$ .

Answer: 67°

Example 2 · Missing leg

A right triangle has hypotenuse 13 and one leg 5. Find the other leg.

Step 1. $b^2 = 13^2 - 5^2 = 169 - 25 = 144$ .
Step 2. $b = 12$ .

Answer: 12

Common mistakes

Adding squares when finding a shorter side (you must subtract).
Mislabelling opposite and adjacent relative to the angle.
Calculator in radians instead of degrees.

Try it yourself

Original practice questions. Your answer is checked instantly.

Question 1Pythagoras and right-triangle trigonometryEasier

A right-angled triangle has legs of length 6 and 8. What is the length of the hypotenuse?

Question 2Pythagoras and right-triangle trigonometryEasier

In a right-angled triangle, which ratio equals $\tan\theta$ ?