Finding a side of a right-angled triangle

The plane and space, grade 11 (3 units) Lesson 9 of 15Learning

Finding a side of a right-angled triangle

Choosing the ratio

When an acute angle and one side of a right-angled triangle are known, any other side can be found:

Mark which side is known and which is wanted: opposite the angle, adjacent to it, or the hypotenuse.
Choose the ratio that holds both: opposite and hypotenuse – sine; adjacent and hypotenuse – cosine; opposite and adjacent – tangent.
Write the ratio as an equation, and get the side on its own.

Example: a ladder against a wall

A ladder 5 m long leans against a wall, and makes an angle of $\displaystyle 70^\circ$ with the ground.
How high up the wall, $\displaystyle h$ , does it reach, and how far from the wall, $\displaystyle d$ , are its feet?

In the solution below the ladder is the hypotenuse; $\displaystyle h$ is opposite the angle, and $\displaystyle d$ next to it.

Opposite and hypotenuse – sine: $\displaystyle {\sin70^\circ}={\frac{h}{5}}$ , so $\displaystyle {h}={5\cdot\sin70^\circ\approx4.70}$ m.
Adjacent and hypotenuse – cosine: $\displaystyle {\cos70^\circ}={\frac{d}{5}}$ , so $\displaystyle {d}={5\cdot\cos70^\circ\approx1.71}$ m.

Exercise 1

A ladder 6 m long leans against a wall, and makes an angle of $\displaystyle 75^\circ$ with the ground.
Round to two decimal places.

a

How high does the ladder reach, in metres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How far from the wall are the ladder’s feet, in metres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

When the side is in the denominator

A kite is 30 m up. Its string is taut and straight, and makes an angle of $\displaystyle 50^\circ$ with the ground.
How long is the string, $\displaystyle x$ ? (The string is held near the ground.)

The height is opposite the angle, and the string is the hypotenuse: $\displaystyle {\sin50^\circ}={\frac{30}{x}}$ .
This time the wanted side is in the denominator. We multiply both sides by $\displaystyle x$ and divide by $\displaystyle \sin50^\circ$ : $\displaystyle {x}={\frac{30}{\sin50^\circ}\approx39.16}$ m.
A check: the string, the hypotenuse, is longer than the height – as it must be.

Exercise 2

Find $\displaystyle x$ , rounded to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

A slide’s ladder reaches a height of 2.5 m, and the slide makes an angle of $\displaystyle 35^\circ$ with the ground.
How long is the slide, in metres? Round to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

A cable is stretched from the top of a mast to the ground.
It is anchored 8 m from the mast’s foot, and makes an angle of $\displaystyle 60^\circ$ with the ground.
Round to two decimal places.

a

How tall is the mast, in metres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

How long is the cable, in metres?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: the plane and space, grade 11 (3 units)

Scale

The real length – $\displaystyle l$ the length on the map, at a scale of $\displaystyle 1:n$
$\displaystyle L$ $\displaystyle =$ $\displaystyle n\cdot l$
The real area – $\displaystyle s$ the area on the map, at a scale of $\displaystyle 1:n$
$\displaystyle S$ $\displaystyle =$ $\displaystyle n^{2}\cdot s$

Similar shapes

The scale factor – $\displaystyle a_1$ , $\displaystyle a_2$ corresponding sides
$\displaystyle k$ $\displaystyle =$ $\displaystyle \frac{a_2}{a_1}$
The ratio of the areas of similar shapes
$\displaystyle \frac{S_2}{S_1}$ $\displaystyle =$ $\displaystyle k^{2}$
A height by its shadow – $\displaystyle H$ , $\displaystyle S$ the object and its shadow; $\displaystyle h$ , $\displaystyle s$ a stick and its shadow
$\displaystyle \frac{H}{S}$ $\displaystyle =$ $\displaystyle \frac{h}{s}$

Trigonometry in the right-angled triangle

Sine – $\displaystyle a$ the side opposite the angle, $\displaystyle c$ the hypotenuse
$\displaystyle \sin\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{c}$
Cosine – $\displaystyle b$ the side adjacent to the angle
$\displaystyle \cos\alpha$ $\displaystyle =$ $\displaystyle \frac{b}{c}$
Tangent – the opposite side over the adjacent side
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{a}{b}$
The angle from its tangent (and likewise $\displaystyle \sin^{-1}$ , $\displaystyle \cos^{-1}$ )
$\displaystyle \alpha$ $\displaystyle =$ $\displaystyle \tan^{-1}\frac{a}{b}$
Pythagoras’ theorem
$\displaystyle a^2+b^2$ $\displaystyle =$ $\displaystyle c^2$
A gradient in percent – $\displaystyle \alpha$ the angle with the horizontal
$\displaystyle p$ $\displaystyle =$ $\displaystyle 100\cdot\tan\alpha$