Rules for right-angled triangles

Trigonometry: Angles with sine, cosine and tangent

Rules for right-angled triangles

There is an important rule for right-angled triangles pertaining the ratio between the sides.

Pythagorean theorem

When it comes to the lengths of the sides of a right-angled triangle with a right angle $\orange C$ , legs $\blue a$ and $\green b$ and hypotenuse (the longest side of a right-angled triangle) $\orange c$ , the following rule applies:

$\blue a^2+\green b^2=\orange c^2$

We call this rule the Pythagorean theorem.

With this theorem, we can calculate the remaining side of a right-angled triangle of which we already know two sides.

For example, if we want to calculate the hypotenuse, we isolate $\orange c$ in the Pythagorean theorem:

$\orange c = \sqrt{\blue a ^2 + \green b^2}$

In a right-angled triangle $ABC$ with right angle $C$ is $AB$ the $\orange{\textbf{hypotenuse}}$ and are $AC$ and $BC$ the legs of the triangle:

$AC$ is the $\green{\textbf{adjacent leg}}$ of $\blue \alpha$
$BC$ is the $\blue{\textbf{opposite leg}}$ of $\blue \alpha$

geogebra plaatje

In addition to the ratio between the sides of a right-angled triangle, there are important relationships between the sides and angles of a right-angled triangle.

Sine, cosine, and tangent in a right-angled triangle,

In a right-angled triangle $\triangle ABC$ with right angle $C$ we define:

$\sin(\alpha)=\frac{\text{opposite side}}{\text{hypotenuse}}=\frac{\blue a}{\orange c}$
$\cos(\alpha)=\frac{\text{adjacent side}}{\text{hypotenuse}}=\frac{\green b}{\orange c}$
$\tan(\alpha)=\frac{\text{opposite side}}{\text{adjacent side}}=\frac{\blue a}{\green b}$

We call $\sin$ (sine), $\cos$ (cosine) and $\tan$ (tangent) trigonometric functions.

With these trigonometric functions we can calculate, using an angle and a side, the remaining sides in a right-angled triangle. We can also calculate the angle using two sides and the inverse.

Calculator

The buttons $\sin$ , $\cos$ and $\tan$ are on your calculator. Note that your calculator is set to degrees instead of radians when working with degrees. We will later have a look at radians, but for now we'll only work with degrees.

The inverses of the trigonometric function $\sin^{-1}$ or $\arcsin$ , $\cos^{-1}$ or $\arccos$ and $\tan^{-1}$ or $\arctan$ are also on your calculator. We can use these to calculate the measure of an angle in a triangle of which we already know two sides.

Other angle

We have now given the expression for the angle $\blue \alpha$ , but in the same way, we can also give expressions for angle $\green \beta$ . In this case, the adjacent side is then equal to side $\blue a$ and the opposite side adjacent to side $\green b$ .

Given triangle $ABC$ :

Calculate side $AC$ exactly.

$AC=$ $5$

$\begin{array}{rcl}AB^2+BC^2&=&AC^2 \\ &&\phantom{xxx}\blue{\text{Pythagorean theorem}} \\ 4^2+3^2&=&AC^2 \\ &&\phantom{xxx}\blue{AB=4 \text{ en } BC=3} \\ 25&=&AC^2 \\ &&\phantom{xxx}\blue{\text{calculated}} \\ \sqrt{25}&=&AC \\ &&\phantom{xxx}\blue{\text{took root on both sides}} \\ AC&=&5 \\ &&\phantom{xxx}\blue{\text{swapped left and right, and simplified where possible}} \end {array}$

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