Pythagoras & Trigonometry

Pythagoras' Theorem

Pythagoras' theorem connects the three sides of any right-angled triangle. If the two shorter sides have lengths $a$ and $b$, and the hypotenuse (the longest side, opposite the right angle) has length $c$, then:

$$a^2 + b^2 = c^2$$

Key terms Hypotenuse — the side opposite the right angle; always the longest side.

Pythagorean triple — three whole numbers that satisfy the theorem, e.g. $(3, 4, 5)$, $(5, 12, 13)$, $(8, 15, 17)$.

Worked example A ladder of length $5$ m leans against a wall, with its foot $1.4$ m from the base. How high up the wall does it reach?

The ladder is the hypotenuse, so $h^2 + 1.4^2 = 5^2$.

Watch out To find a shorter side, subtract: $a^2 = c^2 - b^2$. A common mistake is adding when you should subtract. The hypotenuse must always come out as the largest value.

Pythagoras in 3D

For a cuboid with edges $a$, $b$ and $c$, the space diagonal $d$ (corner to opposite corner) is found by applying Pythagoras twice:

$$d^2 = a^2 + b^2 + c^2$$

Worked example A box measures $6$ cm by $8$ cm by $24$ cm. Find the longest straight pencil that fits inside.

$d^2 = 6^2 + 8^2 + 24^2 = 36 + 64 + 576 = 676$.

Exam tip In 3D problems, sketch the right-angled triangle you actually need. The base diagonal of a cuboid is $\sqrt{a^2 + b^2}$; that diagonal and the vertical height then form a new right-angled triangle whose hypotenuse is the space diagonal.

The Trigonometric Ratios — SOHCAHTOA

For a right-angled triangle with an angle $\theta$, label the sides relative to $\theta$: the opposite, the adjacent, and the hypotenuse. The three ratios are:

$$\sin\theta = \frac{\text{opp}}{\text{hyp}} \qquad \cos\theta = \frac{\text{adj}}{\text{hyp}} \qquad \tan\theta = \frac{\text{opp}}{\text{adj}}$$

Remember them with SOHCAHTOA: Sin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj.

Key terms Opposite — the side facing the angle $\theta$.

Adjacent — the side next to $\theta$ that is not the hypotenuse.

#### Finding a side

Worked example In a right-angled triangle the angle is $35°$ and the hypotenuse is $12$ cm. Find the side opposite the angle.

We have opp and hyp, so use sin: $\sin 35° = \dfrac{x}{12}$.

$x = 12 \times \sin 35° = 12 \times 0.5736 = 6.88$ cm (3 s.f.).

Worked example A right-angled triangle has a $40°$ angle with the adjacent side $9$ cm. Find the opposite side.

Opp and adj means tan: $\tan 40° = \dfrac{x}{9}$.

$x = 9 \times \tan 40° = 9 \times 0.8391 = 7.55$ cm (3 s.f.).

#### Finding an angle

To find an angle, use the inverse functions $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$.

Worked example A right-angled triangle has opposite side $7$ cm and adjacent side $10$ cm. Find $\theta$.

$\tan\theta = \dfrac{7}{10} = 0.7$.

Watch out When the unknown is on the bottom of the fraction (e.g. $\cos 30° = \frac{8}{x}$), rearrange to $x = \frac{8}{\cos 30°}$. Don't multiply when you should divide. Always check your calculator is in degrees mode.

Angles of Elevation and Depression

The angle of elevation is measured upwards from the horizontal to an object above you. The angle of depression is measured downwards from the horizontal to an object below you. These two angles are equal (alternate angles between parallel horizontals).

Worked example From a point $50$ m from the base of a tower, the angle of elevation of the top is $28°$. How tall is the tower?

opp = height, adj = $50$, so $\tan 28° = \dfrac{h}{50}$.

Exact Trigonometric Values

You must know these without a calculator:

Exam tip $\tan 90°$ is undefined. Notice that $\sin$ and $\cos$ are mirror images: $\sin\theta = \cos(90° - \theta)$. Knowing the $30$–$60$–$90$ and $45$–$45$–$90$ triangles lets you reconstruct the whole table.

The Sine Rule

For any triangle (not just right-angled), label each angle with a capital letter and the side opposite it with the matching small letter. The sine rule is:

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

Use it when you have a side and its opposite angle, plus one more side or angle.

Worked example In triangle $ABC$, $A = 40°$, $B = 75°$ and side $a = 8$ cm. Find side $b$.

The Cosine Rule

Use the cosine rule when the sine rule won't work: either you know two sides and the angle between them, or you know all three sides.

$$a^2 = b^2 + c^2 - 2bc\cos A$$

To find an angle, rearrange to:

$$\cos A = \frac{b^2 + c^2 - a^2}{2bc}$$

Worked example A triangle has sides $b = 7$ cm and $c = 10$ cm with the included angle $A = 55°$. Find side $a$.

$a^2 = 7^2 + 10^2 - 2 \times 7 \times 10 \times \cos 55°$.

Worked example A triangle has sides $a = 6$, $b = 8$, $c = 11$. Find the largest angle.

The largest angle faces the longest side, $c$, so: $\cos C = \dfrac{6^2 + 8^2 - 11^2}{2 \times 6 \times 8} = \dfrac{36 + 64 - 121}{96} = \dfrac{-21}{96} = -0.2188$.

Watch out The angle in the cosine rule must be the one between the two given sides. If it isn't, you can't use this version directly — look for the sine rule instead.

Area of a Triangle

When you know two sides and the included angle, the area is:

$$\text{Area} = \frac{1}{2}ab\sin C$$

Worked example Find the area of a triangle with sides $9$ cm and $12$ cm and an included angle of $43°$.

Area $= \frac{1}{2} \times 9 \times 12 \times \sin 43° = 54 \times 0.6820 = 36.8$ cm$^2$ (3 s.f.).

Exam tip Choose your tool by what you're given: right angle → Pythagoras / SOHCAHTOA; opposite pairs → sine rule; two sides + included angle (or three sides) → cosine rule; two sides + included angle for area → $\frac{1}{2}ab\sin C$. Keep full accuracy in your calculator and only round the final answer to 3 s.f.