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Trigonometry Units

Trigonometry is a branch of mathematics that means "measurement of, with and by means of triangles".

A right-angled triangle is a special triangle in which one of the interior angles is a right angle (90°). The longest side of a right-angled triangle is called the hypotenuse.

In all triangles, including right-angled triangles, the sum of the interior angles is 180°.

Pythagoras' Theorem
Pythagoras was a Greek mathematician who lived from 569 -500 B.C.
It is said that he discovered this special property of right-angled triangles while looking at the the tiles of an Egyptian Palace.

"In a right angled triangle, the area of the square on the hypotenuse equals the sum of the squares on the other two sides."

h is the hypotenuse

Examples

Finding the length of the hypotenuse Finding the length of the other side Applications of Pythagoras' Theorem

Example 1
Finding the length of the hypotenuse.

Find the length of the hypotenuse in this right-angled triangle.

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Example 2
Finding the length of other side.

Transpose the formula in Pythagoras' Theorem and use it to find the length of the unknown side (a).

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Example 3

Application of Pythagoras' Theorem
A builder wishes to use corrugated iron (3 metres long) on the gable roof shown.
The vertical height of the gable above D is 1.5 m. What length of rafter (BC) will he need to use?

ABC is an isosceles triangle, so D is the midpoint of BC. We need to calculate BD.
Let BD be x metres. Then by Pythagoras,

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Have a Go

Problem 1

Find the length of the hypotenuse.

Problem 2

A six-metre ladder rests with its foot 1.5 m from a vertical wall.
(a) How high up the wall does it reach?
(b) If its foot is pushed 0.5 m nearer to the wall, how much higher up the wall will it reach?

Practice Questions

Question 1

Find the length of the diagonal of a square
of side 7 cm.

Question 2

Find the side labelled "z" in the following right-angled triangle.

Question 3

A drain pipe is to be laid diagonally across a rectangular block of ground. If the block has a frontage of 88 m and a depth of 182 m, calculate the length of the piping required.

Question 4

What is the longest piece of dowelling
that can be fitted into a rectangular box
21 cm x 16 cm x 12 cm?

Solution 1

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Solution 2
(a)

(b) If the ladder is now 1.0 m from the wall

The ladder will reach (5.92 - 5.81 = 0.11) metres higher up.
Answer: The ladder is approximately 0.1 m higher up.

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