Numeracy - Southern Cross University

Numeracy

Introduction to Trigonometry

Pythagoras' Theorem and basic Trigonometry use right angle triangle structures. (Advanced Trigonometry uses non-right angled triangles)

The angle sum of a triangle is 180?, as one angle is 90? the other two angles must add to 90?.

For this triangle, it is possible to write

+ = 90?

Conventions for naming triangles involve using capital letters for vertices (corners) and lower case letters for sides. You may have already noticed that letters from the Greek Alphabet are used for naming angles.

You will notice that side a is opposite angle A, side

A

b is opposite angle B etc.

b c

Angles can be labelled; (a) using Greek letters = 32

B

a

(b) using the letter at the vertex A =32 C

(c) using three letters BAC = 32

Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources +61 2 6626 9262 | ctl@scu.edu.au | scu.edu.au/teachinglearning

Page 1 [last edited on 7 September 2017]

The three letter method can remove ambiguity in more complex situations.

A

A is ambiguous.

BAC, CAD and BAD are expressed clearly.

Note: BAD = BAC + CAD

B

C

D

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Numeracy

Module contents

? Introduction ? Pythagoras' Theorem ? The Trigonometric Ratios ? Finding Sides ? The Trigonometric Ratios ? Finding Angles ? Answers to activity questions

Outcomes

? To use Pythagoras' Theorem to solve right angled triangle problems. ? To solve right angled triangles, ie: missing sides and missing angles. ? To use Pythagoras' Theorem and trigonometry to solve problems.

Check your skills

This module covers the following concepts, if you can successfully answer these questions, you do not need to do this module. Check your answers from the answer section at the end of the module.

1. (a) Use Pythagoras' Theorem to calculate x. (b) A rectangle has a length of 6.72m and width of 4.83m. Find the length of the diagonal (to 2 d.p.).

x

30 cm

10cm

2. (a) Calculate the value of x (b) Calculate the value of (c) Calculate the value

in this triangle.

x in this triangle.

of in this triangle.

x 40?

12.5m

49? x

7.3km

1.2m

0.85m

3. A kite on the end of a 30m string is flying at an angle of elevation of 72?. What is the height of the kite directly above the ground?

Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources +61 2 6626 9262 | ctl@scu.edu.au | scu.edu.au/teachinglearning

Page 1 [last edited on 7 September 2017]

Numeracy

Topic 1: Pythagoras' Theorem

Pythagoras' Theorem states that in a right angled triangle:

`The square of the hypotenuse is equal to the sum of the squares of the other two sides'

Diagrammatically: a

Hypotenuse c (h)

Other 2 sides

b

The Hypotenuse is the longest side in the triangle and is also opposite the right angle. c=2 a2 + b2

The square of

Sum of the squares of the other

the Hypotenuse =

2 sides

Alternatively, because the Hypotenuse is a unique side: h=2 a2 + b2

This means that Pythagoras' Theorem can be used to find the length of a missing side in a right angled triangle.

Centre for Teaching and Learning | Academic Practice | Academic Skills | Digital Resources +61 2 6626 9262 | ctl@scu.edu.au | scu.edu.au/teachinglearning

Page 1 [last edited on 7 September 2017]

Examples

Hypotenuse unknown

(i)

7

9 h

h2 =a2 + b2 ? rule h2 =72 + 92 ? substitute h=2 49 + 81 h2 = 130

=h 130 ? square root h 11.4

Other side unknown

(i)

27

a 29

h2 =a2 + b2 ? rule

292 = 272 + a2 ? substitute

8= 41 729 + a2

841- 72=9 a2

? rearrange

= 112 a2

? square root

a = 112

a 10.6

Or

h2 =a2 + b2 ? rule

h2 -= b2 a2

? rearrange

292 - 2= 72 a2

? substutite

841- 729 = a2

= 112 a2

? square root

a = 112 a 10.6

(ii) In a right triangle: side a = 5 cm, b = 10 cm and c is the hypotenuse. Determine the length of side c.

h2 =a2 + b2 ? rule h2 =52 +102 ? substitute h=2 25 +100 h2 = 125 =h 125 ? square root h 11.2

(ii)

5 metre ladder

1.2 metres

How far up the wall does the ladder

reach?

h2 =a2 + b2 ? rule

52 = 1.22 + b2 ? substitute

= 25 1.44 + b2

25 -1.4=4 b2

? rearrange

= 23.56 b2

? square root

b = 23.56

b 4.85

The ladder reaches approximately 4.85m up the wall.

Mixed worded questions

(i) A shed has a gable roof as drawn below. Calculate the length of sheets of roofing iron required in its construction.

l

1.6m

4.8 m

Let the length of the sheets be l.

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2.4 m

h=2 a2 + b2 =l 2 1.62 + 2.42 =l2 2.56 + 5.76 l2 = 8.32

l = 8.32 l = 2.88 The length of the sheets of roofing iron is

2.88m

(ii) A guy (support) wire is attached 3.2 m up a pole and at a point 2.1 m from the pole. The ground and the pole are perpendicular (at right angles). What is the length of the guy wire?

Let the length of the guy wire be l. h=2 a2 + b2

=l2 3.22 + 2.12 l2 = 14.65

l = 14.65 l = 3.83 The length of the guy wire is 3.83m

Guy wire Pole

Ground

(iii) On a softball diamond, the distance between bases is 60 feet or 18.29m. How far must the catcher (at home base) throw the ball to the player on second base?

Second Base

18.29m

18.29m

First Base

The angle at first base is a right angle. Let the distance from Home plate to second base be d.

h=2 a2 + b2

= d 2 18.292 +18.292

d 2 = 669.05

d = 669.05

d = 25.87 The distance from home plate to second

base is 25.87m

Home Plate

Pythagorean Triple or Triad

Sometimes, the three lengths of a right angled triangle are all whole numbers. When this occurs, they are called a Pythagorean Triple or Triad. The most commonly known of these is (3,4,5) representing the triangle below:

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3

5

4 Three other Triads are (5,12,13), (7,24,25) and (8,15,17).

h=2 a2 + b2 5=2 32 + 42 25= 9 +16 25 = 25

Other Triads can be based on multiples of the base Triads; the Triads (6,8,10), (9,12,15), (12,16,20) ... are based the base Triad (3,4,5).

If it is important to decide if a triangle is a right angled triangle, then Pythagoras' Theorem can be used to decide this.

15.4

9.8

19.8

= h2 1= 9.82 392.04 a2 + b2= 15.42 + 9.82 = 237.16 + 96.04

= 333.2 h2 a2 + b2

This is not a right angled triangle.

4 10.4

9.6

= h2 1= 0.42 108.16 a2 + b2 = 42 + 9.62

= 16 + 92.16 = 108.16

h=2 a2 + b2

This is a right angled triangle.

Video `Pythagoras' Theorem'

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Activity

1. Use Pythagoras' Theorem to find the missing length.

(a)

(b)

x

x 26

18 cm

12.5 cm

20

(c) Find the hypotenuse when a=6.1 and b=3.4

(e)

(d) Find the missing side given a=23.5 and h=40

(f)

6 m x

6 m 8 m

h

10 mm

Find the height of the triangle. Also calculate the area.

(g) A orienteering participant runs (h)

**A Real Challenge**

650m north an then turns and runs 1.4 km east. How far from the starting point is the runner?

A square has a diagonal of 20 cm, what is the side length?

2. Do the following triangles contain a right angle?

(a) 10, 24, 26

(b) 7, 8, 10

(d) 1.4, 4.8, 5

(e) 6, 6, 8

(c) 2, 4.8, 5.2 (f) 5, 6, 7

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