The trigonometric functions in a right-angled triangle

From this lesson to the end of the course the angles in triangles are measured in degrees: $\displaystyle {\sin30^{\circ}}={\frac{1}{2}}$, like $\displaystyle {\sin\frac{\pi}{6}}={\frac{1}{2}}$. The calculator is set to degrees (DEG).

Everything learnt on the unit circle still holds, only the writing changes: $\displaystyle \pi$ radians are $\displaystyle 180^{\circ}$, so, for example, $\displaystyle {\sin(180^{\circ}-\alpha)}={\sin\alpha}$ and $\displaystyle {\cos(180^{\circ}-\alpha)}={-\cos\alpha}$.

On the board: a right-angled triangle, the acute angle $\displaystyle \alpha$ at $\displaystyle A$, the leg $\displaystyle a$ opposite it, the leg $\displaystyle b$ beside it and the hypotenuse $\displaystyle c$.

Shrink the triangle $\displaystyle c$ times: the hypotenuse becomes of length $\displaystyle 1$, and with $\displaystyle A$ at the origin and $\displaystyle b$ on the $\displaystyle x$-axis, the vertex $\displaystyle B$ is on the unit circle, at the point $\displaystyle (\cos\alpha,\sin\alpha)$. So the legs of the shrunk triangle are $\displaystyle \cos\alpha$ and $\displaystyle \sin\alpha$, and those of the triangle itself – $\displaystyle c$ times as long:

• sine – the leg opposite the angle over the hypotenuse;
• cosine – the leg beside the angle over the hypotenuse;
• tangent – the leg opposite the angle over the leg beside it.

The example below works out both legs of a right-angled triangle whose hypotenuse is $\displaystyle 10$ and one of whose angles is $\displaystyle 35^{\circ}$.

• The leg opposite the angle: $\displaystyle {a}={10\sin35^{\circ}\approx5.74}$;
• the leg beside the angle: $\displaystyle {b}={10\cos35^{\circ}\approx8.19}$.

And the other way round – an angle from the sides: if the legs are $\displaystyle 5$ and $\displaystyle 12$, then $\displaystyle {\tan\alpha}={\frac{5}{12}}$, and $\displaystyle {\alpha}={\tan^{-1}\frac{5}{12}\approx22.62^{\circ}}$.

• In working out a real length an approximate value is enough: $\displaystyle \sin45^{\circ}\approx0.707$.
• In a proof, and in any result you go on calculating with, use the exact value: $\displaystyle {\sin45^{\circ}}={\frac{\sqrt{2}}{2}}$.

Round only at the end of the calculation: rounding in the middle adds an error to every step after it.

In triangle $\displaystyle ABC$ the angle $\displaystyle C$ is a right angle.
$\displaystyle {AB}={12}$, and $\displaystyle {\angle A}={30^{\circ}}$.
Work out the exact value.

The legs of a right-angled triangle are $\displaystyle 5$ and $\displaystyle 12$.
Work out the angle opposite the leg $\displaystyle 5$, in degrees.
Round to two decimal places (the last line starts with $\displaystyle \approx$).

In the isosceles triangle $\displaystyle ABC$: $\displaystyle {AB}={AC}={10}$, and the base angle $\displaystyle {\angle B}={70^{\circ}}$.
$\displaystyle AD$ is the height to the base.
Round to two decimal places.

In a right-angled triangle the hypotenuse is $\displaystyle c$ and one of the angles is $\displaystyle \alpha$.
Express both legs and the area of the triangle in $\displaystyle c$ and $\displaystyle \alpha$.