Practice: trigonometry

The plane and space, grade 11 (3 units) Lesson 14 of 15Practice

Practice: trigonometry

In every question, round to two decimal places.

1

Find $\displaystyle x$ .

/ – 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

2

Find the angle $\displaystyle \alpha$ .

/ – 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

3

A ladder 7 m long reaches 6.5 m up a wall.
What is the angle between the ladder and the ground?

/ – 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

4

From 25 m away from a pylon, the angle of elevation to its top is $\displaystyle 60^\circ$ .
The surveyor’s eyes are 1.5 m above the ground.
How tall is the pylon, 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

5

From the top of a cliff 120 m high, a boat is seen at an angle of depression of $\displaystyle 20^\circ$ .
How far is the boat from the foot of the cliff, 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

6

A road’s gradient is 15%.

a

What is the angle between the road and the horizontal?

/ – 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 many metres do you climb driving 3 km along the road?

/ – 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

7

The front of a roof is an isosceles triangle.
Its base is 10 m, and the roof is 3 m high.
What are its base angles?

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