Trigonometry with right-angled triangles

Sine, cosine and tangent in a right-angled triangle, the exact values for 30°, 45° and 60°, the reciprocal ratios, and finding unknown sides and angles.

The three ratios

Label the sides relative to the angle $\theta$: the hypotenuse (opposite the right angle), the opposite side and the adjacent side.

SOH CAH TOA

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

Similar triangles have the same ratios, which is why each ratio depends only on the angle.

A right-angled triangle with adjacent side 5

The triangle has its angle θ at the origin, adjacent side 5 along the x-axis and the opposite side on x = 5. Change θ and watch the ratios.

Try this

Set $\theta = 45^\circ$. Why is $\tan 45^\circ = 1$?
What happens to $\cos\theta$ as $\theta$ gets close to $90^\circ$?

Exact values

Two triangles give the exact values: half a square (sides $1, 1, \sqrt{2}$) and half an equilateral triangle (sides $1, \sqrt{3}, 2$).

$\theta$	$30^\circ$	$45^\circ$	$60^\circ$
$\sin\theta$	$\tfrac12$	$\tfrac{1}{\sqrt{2}}$	$\tfrac{\sqrt{3}}{2}$
$\cos\theta$	$\tfrac{\sqrt{3}}{2}$	$\tfrac{1}{\sqrt{2}}$	$\tfrac12$
$\tan\theta$	$\tfrac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$

The reciprocal ratios

Reciprocal ratios

$$\operatorname{cosec}\theta = \frac{1}{\sin\theta} \qquad \sec\theta = \frac{1}{\cos\theta} \qquad \cot\theta = \frac{1}{\tan\theta}$$

Finding sides and angles

Example 1: a side●

In a right-angled triangle, the hypotenuse is $12$ and one angle is $35^\circ$. Find the side opposite $35^\circ$ to 1 decimal place.

$\sin 35^\circ = \dfrac{x}{12}$

Opposite and hypotenuse: use sine.

$x = 12\sin 35^\circ \approx 6.9$

Example 2: an angle●

The opposite side is $7$ and the adjacent side is $4$. Find $\theta$ to the nearest degree.

$\tan\theta = \dfrac{7}{4}$

$\theta = \tan^{-1}\left(\dfrac{7}{4}\right) \approx 60^\circ$

Use the inverse key, tan⁻¹.

Flashcards

SOH CAH TOA

$\sin = \dfrac{o}{h}$, $\cos = \dfrac{a}{h}$, $\tan = \dfrac{o}{a}$

sin 30°, cos 30°, tan 30°

$\tfrac12$, $\tfrac{\sqrt{3}}{2}$, $\tfrac{1}{\sqrt{3}}$

sin 45°

$\tfrac{1}{\sqrt{2}}$

sec θ

$\dfrac{1}{\cos\theta}$

Practice

$\cos 60^\circ =$
$\tan 30^\circ =$
Find the exact value of $\sec 60^\circ$.
The adjacent side is $10$ and the angle is $40^\circ$. Find the opposite side to 1 decimal place.
$\sin\theta = 0.5$ and $\theta$ is acute. Find $\theta$ in degrees.
The ratio $\dfrac{\text{adjacent}}{\text{opposite}}$ is:
Find the exact length of the hypotenuse of a right-angled isosceles triangle with legs of length $1$. (Type something like sqrt(5).)
Which is largest?