This note covers Pythagoras' theorem and the sine, cosine and tangent ratios in right-angled triangles. Together they let you find missing sides and angles, including in problems about ramps, ladders, heights and distances. At Extended level you also use exact values of the ratios for some special angles.
Key ideas
In a right-angled triangle, the longest side is called the hypotenuse. It is opposite the right angle.
Pythagoras' theorem. The square on the hypotenuse equals the sum of the squares on the other two sides. If the hypotenuse is \(c\) and the other sides are \(a\) and \(b\), then \(c^2 = a^2 + b^2\).
- To find the hypotenuse, add the squares of the two shorter sides, then take the square root.
- To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root.
- The theorem works the other way too. If three sides satisfy \(a^2 + b^2 = c^2\), the triangle has a right angle. For example, 5, 12, 13 gives \(25 + 144 = 169\).
- Pythagoras also gives the distance between two points on a grid, by making a right-angled triangle from the horizontal and vertical differences.
Trigonometric ratios. Pick an angle \(\theta\) (not the right angle). Label the sides from that angle's point of view:
- Hypotenuse: the longest side.
- Opposite: the side across from \(\theta\).
- Adjacent: the side next to \(\theta\) that is not the hypotenuse.
The three ratios are \(\sin\theta = \dfrac{\text{opp}}{\text{hyp}}\), \(\cos\theta = \dfrac{\text{adj}}{\text{hyp}}\) and \(\tan\theta = \dfrac{\text{opp}}{\text{adj}}\). The memory aid SOH CAH TOA helps you recall them.
- To find a side, choose the ratio that links the side you want with the side you know. Substitute and rearrange.
- To find an angle, use the inverse function on your calculator: \(\sin^{-1}\), \(\cos^{-1}\) or \(\tan^{-1}\).
- Make sure your calculator is in degree mode.
Angles of elevation and depression. The angle of elevation is the angle measured upwards from the horizontal to a line of sight. The angle of depression is measured downwards from the horizontal. Draw the horizontal line first, then build the right-angled triangle.
Exact values (Extended). For some angles, the ratios can be written exactly without a calculator:
- \(\sin 30^\circ = \tfrac{1}{2}\), \(\cos 30^\circ = \tfrac{\sqrt{3}}{2}\), \(\tan 30^\circ = \tfrac{1}{\sqrt{3}}\).
- \(\sin 45^\circ = \cos 45^\circ = \tfrac{\sqrt{2}}{2}\), \(\tan 45^\circ = 1\).
- \(\sin 60^\circ = \tfrac{\sqrt{3}}{2}\), \(\cos 60^\circ = \tfrac{1}{2}\), \(\tan 60^\circ = \sqrt{3}\).
Choosing between the three ratios is the main skill. Ask which two sides are involved: the one you know and the one you want. If they are opposite and hypotenuse, use sine. If they are adjacent and hypotenuse, use cosine. If they are opposite and adjacent, use tangent. Many problems also need a sketch first, because a clear right-angled triangle makes the labels obvious.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \(c^2 = a^2 + b^2\) | Pythagoras: \(c\) is the hypotenuse. Use when two sides are known and the third is wanted. |
| \(\sin\theta = \dfrac{\text{opp}}{\text{hyp}}\) | Use when the opposite side and the hypotenuse are involved. |
| \(\cos\theta = \dfrac{\text{adj}}{\text{hyp}}\) | Use when the adjacent side and the hypotenuse are involved. |
| \(\tan\theta = \dfrac{\text{opp}}{\text{adj}}\) | Use when the opposite and adjacent sides are involved. |
| \(\theta = \sin^{-1}\!\left(\dfrac{\text{opp}}{\text{hyp}}\right)\) | Finding an angle from two known sides. The same idea applies to \(\cos^{-1}\) and \(\tan^{-1}\). |
| Angle of elevation / depression | Measured up / down from the horizontal. |
Worked examples
Example 1 A ladder 6.5 m long leans against a vertical wall. Its foot is 2.5 m from the base of the wall. How high up the wall does it reach?
Step 1: the ladder is the hypotenuse, 6.5 m. The distance along the ground is 2.5 m.
Step 2: height\(^2 = 6.5^2 - 2.5^2 = 42.25 - 6.25 = 36\).
Step 3: height \(= \sqrt{36} = 6\) m.
Answer: 6 m
Example 2 A wheelchair ramp is 12 m long and rises 1.5 m. Find the angle the ramp makes with the horizontal.
Step 1: the ramp is the hypotenuse (12 m). The rise is opposite the angle \(\theta\) (1.5 m).
Step 2: opposite and hypotenuse give sine. \(\sin\theta = \dfrac{1.5}{12} = 0.125\).
Step 3: use the inverse sine function on your calculator. \(\theta = \sin^{-1}(0.125) = 7.18^\circ\).
Answer: \(7.18^\circ\) (3 s.f.)
Example 3 From a point on level ground 40 m from the base of a tower, the angle of elevation of the top of the tower is \(28^\circ\). Find the height of the tower.
Step 1: the distance on the ground is adjacent (40 m). The height is opposite. Opposite and adjacent give tangent.
Step 2: \(\tan 28^\circ = \dfrac{h}{40}\), so \(h = 40 \times \tan 28^\circ\).
Step 3: use the tangent function on your calculator. \(\tan 28^\circ = 0.5317\), so \(h = 21.27\).
Answer: about 21.3 m (3 s.f.)
Common errors
- Using \(c^2 = a^2 + b^2\) with the hypotenuse in the wrong place. When finding a shorter side, you must subtract.
- Forgetting to take the square root at the end of a Pythagoras calculation.
- Labelling opposite and adjacent from the wrong angle. The labels change when you move to the other acute angle.
- Having the calculator in radian or gradian mode instead of degrees.
- Dividing the wrong way when the unknown is in the denominator, for example writing \(x = 9 \times \cos 52^\circ\) instead of \(x = \dfrac{9}{\cos 52^\circ}\).
Exam tip: label the three sides hyp, opp and adj on the diagram before choosing a ratio. Write the equation with the numbers in it, then rearrange. Keep extra decimal places during working and round only at the end.
Check yourself
- Is a triangle with sides 7 cm, 24 cm and 25 cm right-angled?
Answer
\(7^2 + 24^2 = 49 + 576 = 625 = 25^2\). Yes, it is right-angled, with the right angle opposite the 25 cm side. - In a right-angled triangle, the side opposite \(\theta\) is 8 cm and the side adjacent is 15 cm. Find \(\theta\) and the hypotenuse.
Answer
\(\tan\theta = \dfrac{8}{15}\), so \(\theta = \tan^{-1}\!\left(\dfrac{8}{15}\right) \approx 28.1^\circ\). The hypotenuse is \(\sqrt{8^2 + 15^2} = \sqrt{289} = 17\) cm. - A right-angled triangle has hypotenuse 12 cm and an angle of \(35^\circ\). Find the side opposite that angle.
Answer
\(\sin 35^\circ = \dfrac{x}{12}\), so \(x = 12 \sin 35^\circ \approx 6.88\) cm. - (Extended) A right-angled triangle has an angle of \(30^\circ\) and hypotenuse 10 cm. Find the opposite side exactly.
Answer
\(\sin 30^\circ = \tfrac{1}{2}\), so the opposite side is \(10 \times \tfrac{1}{2} = 5\) cm.