The isosceles triangle and the rectangle in trigonometry

The plane and space, grade 11 (3 units) Lesson 13 of 15Learning

The isosceles triangle and the rectangle in trigonometry

The isosceles triangle

The height to the base of an isosceles triangle is also the median to the base and the bisector of the apex angle. So it splits the triangle into two congruent right-angled triangles, and in each of them:

the hypotenuse is a leg of the isosceles triangle;
one side is half the base;
the other side is the height.

Example: the front of a roof

The front of a tiled roof is an isosceles triangle.
Its base is 8 m, and its base angles are $\displaystyle 30^\circ$ .
How high is the roof, $\displaystyle h$ , and how long is a sloping side, $\displaystyle x$ ?

The solution below works in the right-angled triangle $\displaystyle ABD$ , where $\displaystyle BD$ is half the base: 4 m.

$\displaystyle h$ is opposite the angle, and 4 next to it: $\displaystyle {\tan30^\circ}={\frac{h}{4}}$ , so $\displaystyle {h}={4\cdot\tan30^\circ\approx2.31}$ m.
4 is next to the angle, and $\displaystyle x$ is the hypotenuse: $\displaystyle {\cos30^\circ}={\frac{4}{x}}$ , so $\displaystyle {x}={\frac{4}{\cos30^\circ}\approx4.62}$ m.

Exercise 1

A tent’s doorway is an isosceles triangle.
Each sloping side is 2 m long, and the base angles are $\displaystyle 55^\circ$ .
Round to two decimal places.

a

How tall is the tent, 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 wide is the base, 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

Exercise 2

A stepladder has two legs, each 1.5 m long.
The ends of its legs are 1.2 m apart on the floor.
What is the angle between the two legs at the top of the ladder? 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

The rectangle and its diagonal

A diagonal splits a rectangle into two congruent right-angled triangles: the rectangle’s sides are the legs, and the diagonal is the hypotenuse.
A television screen is 120 cm wide and 67.5 cm high:

The angle between the diagonal and the bottom side: $\displaystyle {\tan\alpha}={\frac{67.5}{120}}$ , so $\displaystyle \alpha\approx29.36^\circ$ .
The diagonal’s length, by Pythagoras’ theorem: $\displaystyle \sqrt{120^2+67.5^2}\approx137.68$ cm.
A screen’s size is measured in inches along its diagonal. An inch is 2.54 cm, so this is a screen of about 54 inches.

Exercise 3

A rectangular football pitch is 80 m long and 50 m wide.
What is the angle between its diagonal and its long side? 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 rectangular gate is 3 m wide.
A brace runs along its diagonal, at an angle of $\displaystyle 35^\circ$ to the bottom side.
Round to two decimal places.

a

How tall is the gate, 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 brace, 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

Exercise 5

A warning sign is an equilateral triangle with sides of 90 cm.
Round to two decimal places.

a

How tall is the triangle, in cm?

/ – 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

What is the sign’s area, in cm²?

/ – 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
$\displaystyle p$ $\displaystyle =$ $\displaystyle 100\cdot\tan\alpha$