The six trigonometric ratios

Three ratios of the sides

Take a right triangle $ABC$ with the right angle at $B$, and look from angle $A$. Three sides give six fractions. The first three have names.

Say them as sine, cosine and tangent of $A$. Many students remember them with the words SOH CAH TOA: Sine is Opposite over Hypotenuse, and so on.

Three more: the reciprocals

Turn each fraction upside down and you get the other three ratios.

So $\operatorname{cosec} A = \dfrac{1}{\sin A}$, $\sec A = \dfrac{1}{\cos A}$ and $\cot A = \dfrac{1}{\tan A}$.

Divide sine by cosine and the hypotenuse cancels:

In the same way, $\cot A = \dfrac{\cos A}{\sin A}$.

A note on writing. $\sin A$ is one number: the sine of angle $A$. It is not "sin" times $A$. The word on its own means nothing.

A worked example

In a right triangle, the side opposite angle $A$ is $8\ \text{cm}$, the adjacent side is $15\ \text{cm}$ and the hypotenuse is $17\ \text{cm}$. Write all six ratios.

1. The first three. $\sin A = \dfrac{8}{17}$, $\cos A = \dfrac{15}{17}$, $\tan A = \dfrac{8}{15}$.
2. The reciprocals. $\operatorname{cosec} A = \dfrac{17}{8}$, $\sec A = \dfrac{17}{15}$, $\cot A = \dfrac{15}{8}$.
3. Check. $\dfrac{\sin A}{\cos A} = \dfrac{8}{17} \times \dfrac{17}{15} = \dfrac{8}{15}$, which is $\tan A$.

The size of the triangle does not matter

Draw two right triangles with the same angle $A$, one small and one large. They are similar, so their sides are in the same proportion. Every side of the large one is the same multiple of the small one, and that multiple cancels in each fraction.

So a trigonometric ratio depends only on the angle. That is why we can speak of "$\sin 30°$" without saying which triangle.

How big can a ratio be?

The hypotenuse is the longest side. So for an acute angle:

• $\sin A$ and $\cos A$ are always less than $1$.
• $\operatorname{cosec} A$ and $\sec A$ are always greater than $1$.
• $\tan A$ and $\cot A$ can be any positive number.

Practise with new numbers

Try it in Python

This program writes the six ratios of the $8$, $15$, $17$ triangle as fractions and decimals. It then makes every side three times as long.

Common mistakes

• Mixing up opposite and adjacent. Always name the sides from the angle in the question first.
• Pairing the wrong reciprocals. $\operatorname{cosec}$ goes with $\sin$, and $\sec$ goes with $\cos$.
• Writing $\sin$ without an angle. $\sin A$ is a number; "sin" alone is not.
• Thinking a larger triangle has a larger sine. The ratio depends on the angle only.

Remember these

• $\sin A = \dfrac{\text{opp}}{\text{hyp}}$, $\cos A = \dfrac{\text{adj}}{\text{hyp}}$, $\tan A = \dfrac{\text{opp}}{\text{adj}}$.
• $\operatorname{cosec} A$, $\sec A$ and $\cot A$ are their reciprocals.
• $\tan A = \dfrac{\sin A}{\cos A}$ and $\cot A = \dfrac{\cos A}{\sin A}$.
• A ratio depends only on the angle. For an acute angle, $\sin A < 1$ and $\cos A < 1$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. In a right triangle the opposite side of angle $B$ is $5\ \text{cm}$ and the hypotenuse is $13\ \text{cm}$. What is $\sin B$?

2. If $\tan A = \dfrac{3}{4}$, what is $\cot A$?

3. Which ratio is the reciprocal of $\cos A$?

4. Mohan says $\sin A = \dfrac{6}{5}$ for some acute angle $A$. Can he be right?

5. Two right triangles have the same angle $P$, but one is twice as large. How do their values of $\cos P$ compare?