The trigonometric ratios

Labelling the hypotenuse, opposite and adjacent sides, defining sine, cosine and tangent, and writing ratios for a given angle.

Labelling sides

For an angle $\theta$: the hypotenuse is opposite the right angle, the opposite side faces $\theta$, and the adjacent side is next to $\theta$ (not the hypotenuse).

The three ratios

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

Ratios change with the angle

For a fixed angle the ratios are always the same, whatever the size of the triangle. That is why trigonometry works.

Example 1●

A right-angled triangle has sides $3$, $4$ and $5$. Angle $\theta$ is opposite the side of length $3$. Write the three ratios.

$O = 3$, $A = 4$, $H = 5$

$\sin\theta = \dfrac{3}{5}$, $\cos\theta = \dfrac{4}{5}$, $\tan\theta = \dfrac{3}{4}$

Flashcards

SOH

$\sin = O/H$

CAH

$\cos = A/H$

TOA

$\tan = O/A$

Practice

Sides $3, 4, 5$; $\theta$ is opposite $3$. Find $\sin\theta$.
Same triangle. Find $\cos\theta$.
Same triangle. Find $\tan\theta$.
Same triangle. Let $\alpha$ be the other acute angle. Find $\tan\alpha$.
Sides $5, 12, 13$; $\theta$ is opposite $12$. Find $\sin\theta$.
Same triangle. Find $\cos\theta$.
Which side is never labelled opposite or adjacent?
Use the graph or a calculator: $\sin 30^\circ = $ ?