Sine, cosine and tangent

In a right-angled triangle, the hypotenuse lies opposite the right angle – the longest side. Each of the two acute angles has:

• the opposite side – the leg that does not touch it;
• the adjacent side – the leg that is one of the two sides of the angle.

In the drawing, for the angle $\displaystyle \alpha$: $\displaystyle a$ is the side opposite $\displaystyle \alpha$, $\displaystyle b$ the side adjacent to $\displaystyle \alpha$, and $\displaystyle c$ the hypotenuse. For the angle $\displaystyle B$ the roles swap: $\displaystyle b$ is opposite it, and $\displaystyle a$ adjacent to it.

All right-angled triangles with the same acute angle $\displaystyle \alpha$ are similar to each other (AA): each has a right angle and the angle $\displaystyle \alpha$. So the ratio of two sides is the same in all of them – it depends only on the angle, not on the triangle’s size. For example, in every right-angled triangle with an angle of $\displaystyle 30^\circ$, the side opposite that angle is half the hypotenuse.

The three ratios have names:

• sine – the opposite side over the hypotenuse:

• cosine – the adjacent side over the hypotenuse:

• tangent – the opposite side over the adjacent side:

A calculator gives the sine, cosine and tangent of any angle – when it is set to degrees (DEG on its screen).

• $\displaystyle {\sin30^\circ}={0.5}$, exactly.
• $\displaystyle {\tan40^\circ}={0.8390996\ldots}$ – a number that never ends. We round it, for example to two decimal places, and write $\displaystyle \tan40^\circ\approx0.84$.
• The sign $\displaystyle \approx$ means “is approximately equal to”: the number after it is a rounding of what comes before it.

Calculate, and round to two decimal places.

In the right-angled triangle in the drawing, the legs are 3 and 4, and the hypotenuse is 5:

• $\displaystyle {\sin\alpha}={\frac{3}{5}}={0.6}$: the side opposite $\displaystyle \alpha$ is 3.
• $\displaystyle {\cos\alpha}={\frac{4}{5}}={0.8}$: the side adjacent to $\displaystyle \alpha$ is 4.
• $\displaystyle {\tan\alpha}={\frac{3}{4}}={0.75}$.

In the right-angled triangle in the drawing, the legs are 5 and 12, and the hypotenuse is 13.
Write the ratio, and round to two decimal places.

Why is the sine of an acute angle in a right-angled triangle always less than 1?
Is the tangent always less than 1 too?
Explain.