16

           
 April 2001

Number 16

Maths for Physics: Trigonometry 1 This Factsheet will review the use of:
Pythagoras’ Theorem and Trigonometry in right-angled triangles
Sine and cosine rule for other triangles
Correct use of calculators in trigonometric calculations

2. Trigonometry in right-angled triangles Labelling triangles To use trigonometry, you need to be able to label the sides of a triangle. This depends on which angle you are interested in (i.e. which angle you know, or need to find)
The hypotenuse (H) is the longest side, opposite the right angle
The adjacent (A) is the side next to the angle you are looking at
The opposite (O) is the side opposite the angle you are looking at

1. Pythagoras’ Theorem For a right-angled triangle shortest side2 + middle side2 = longest side2

So which side is the adjacent and which the opposite depends on the angle you are interested in, but the hypotenuse doesn’t change. O

The longest side (or hypotenuse) is always the one opposite the right angle.

O

longest side (hypotenuse)

H A

Example 1. Find the side BC in the triangle below A 2 cm B

C

The calculator does sin, cos or tan before multiplication or division. So if you want to find 2  sin30, for example, you’d type in 2  SIN 30 or 2  30 SIN (depending on your calculator). The answer should be 1.

Example 2. Find the side PQ in the triangle below R 2 cm

Q

If, instead, you wanted to find sin (2  30), you’d need to type in SIN (2  30) or (2  30) SIN (depending on your calculator). The answer should be 0.866…

P

Method 1. longest side or shorter side? 2. For a longer side, work out

Angle we are interested in

Using your calculator with sin, cos and tan If you want to find the sin, cos or tan of an angle on your calculator, you will either have to do SIN 30 or 30 SIN Check which it is now – you should get 0.5 as the answer.

(4 2  2 2 ) = 3.46cm (3SF)

(longest side 2  other side 2 )

4cm

H

Some people find this helps them to remember SOHCAHTOA: “Shouting Often Helps Calm Angry, Heated Teachers Or Adults” There are plenty of other ways to remember it – make up your own!

we want BC, which is one of the shorter sides

2. For a shorter side, work out

A

Sine, cosine and tangent You need to remember how sin, cos and tan relate to the sides of a triangle. One way to do this is to use SOHCAHTOA This stands for Sin = O H Cos = A H Tan = O A

4cm

Method 1. Decide whether you want the longest side or one of the shorter ones

Angle we are interested in

QP is the longest side You will also need to find sin-1, cos-1 and tan-1 on your calculator – these are the “opposites” of sin, cos and tan. You get to them by pressing INV (or, on some calculators 2ND or SHIFT), then SIN, COS or TAN.

(4 2  2 2 ) = 4.47cm (3SF)

(shortest side 2  middle side 2 )

These are used for finding the angle when you know its sin, cos or tan. e.g: if you know cosx = 0.5, then to find x, you press INV COS 0.5 or 0.5 INV COS (depending on your calculator) You should get the answer 60o

Tips: 1. Remember: add for the longest side, subtract for a shorter side 2. Do not attempt to use Pythagoras unless your triangle is definitely right-angled.

1


   

Trigonometry 1  Tip: Many students lose marks through having their calculator in the wrong mode. Calculators have three modes for angles – D (degrees), R(radians) and G (grads). You should always check you are in the correct mode before starting any calculation. All of this Factsheet will involve degrees only (although you will need to use radians in some A2 work).

Example 3. Find angle x in the triangle below 4 cm x 8 cm 4 cm A x

1. Label

Using trigonometry in problems The following three worked examples illustrate the method.

8 cm H

Example 1. Find side AB in the triangle below B

2. SOHCAHTOA

We’ve got A and H  want CAH

3. Formula

cos = A

4. Put in what you know

cosx = 4

5. If you’re trying to find an angle, work out the division then use INV

cosx = 4  8 = 0.5 x = cos-10.5 = INV COS 0.5 = 60o

8 cm 20o

A

B

Method 1. Label only the sides you know, or have to find

H

O

8 cm 20o

A

We’re looking for O and H So we want SOH

3. Write down the formula

sin = O

4. Put in what you know

sin 20o = AB

5. If what you want to find is on the top of the fraction, you need to multiply

8
sin20o = AB 2.74 cm = AB (3 SF)

How do I know whether to use Pythagoras or Trigonometry?
If you know two sides and are not asked about angles, use Pythagoras
If you only know one side, or you have to find an angle, use trigonometry
If you could use either (for example, you know two sides and an angle) then it doesn’t matter which you choose, but since Pythagoras is simpler, it’s advisable to use it.

H 8

Before moving on to the next section, you should attempt questions 1,2 and 3 at the end of the Factsheet to ensure you are happy with trigonometry in right-angled triangles.

Example 2. Find PQ in the triangle below P

3. Trigonometry in other triangles – sine and cosine rule

o

The sine and cosine rule allow you to use trigonometry in any triangle. You may find them useful in problems involving forces and vectors. P

R

8

C

2. Look at SOHCAHTOA to find where the two sides you’ve labelled occur together

50

H

C

5 cm

Q

50

A

1. Label R

3. Formula 4. Put in what you know

5. If what you want is on the bottom of the fraction, you need to swap

Q

5 cm

O 2. Look at SOHCAHTOA

Labelling triangles When using the sine and cosine rule, you need to use the labelling described below, since the labelling used for right-angled triangles no longer works.

o

This is how it works
Capital letters stand for angles
The corresponding lower-case letter stands for the side opposite the angle (so side a is opposite angle A)

We have O and A So we need TOA tan = O

c

A

tan50o = 5 tan50o = 5 PQ = 5

A b C a

PQ

B

PQ

tan50 o

Tip: If the triangle in the question is not labelled in this way, you will find it helpful to re-label it like this before starting any calculations.

= 4.20 cm (3SF)

2


   

Trigonometry 1  The sine rule

The cosine rule

The sine rule says:

The cosine rule says:

sinA sinB sinC   a b c

a2 = b2 + c2 – 2bc cosA

or

b2 = a2 + c2 – 2ac cosB c2 = a2 + b2 – 2ab cosC

or or

a b c   sinA sinB sinC

The examples below show how to use the cosine rule. The examples below show how to use the sine rule.

Example 1. Find the side BC in the triangle below B

Example 1. Find the side BC in the triangle below 5cm C 100o

30o 20o

A

A

8 cm

B

a

Method 1. Relabel the triangle

2. Decide “which way up” to use the sine rule – if you want a side, use it with sides on top; if an angle, have sinA etc on top

b 20o

a sin20 o a

4. Cross out the term that only has letters in it.

sin20

5. Rearrange to find the unknown side

o



b 8  sinB sin100 o



b 8  sinB sin100 o

B

8 o  sin20 sin100 o = 2.78 cm (3SF)

a =

A

2. By looking at the angle you know or are asked to find, decide which form of the cosine rule to use

We know angle A So we use the form with A in it a2 = b2 + c2 – 2bc cosA

3. Substitute in

a2 = 82 + 52 – 2(8)(5)cos300 a2 = 64 + 25 – 80cos30o = 19.7… a = 19.7… = 4.44cm (3 SF)

C

Method

a

1. Relabel B

C 120o

5 cm b

c 6 cm

8 cm b

Example 2. Find the angle C in the triangle below A 6cm 5 cm

5 cm

6 cm

A

Tips: 1. Remember BIDMAS – you must work out 2
b
c
cosA before subtracting it from b2 + c2 2. Do not write down and use rounded figures for a2 or cosA in examples like this – keep the full answer in your calculator. Learning to use the memory on the calculator will help.

Example 2. Find the angle B in the triangle below C 120o

a C

30o

A

We want a side – so use a b c   sinA sinB sinC

3. Substitute in what you know

B

Method 1. Relabel the triangle

8 cm

c

8 cm c 5cm

C 100o

B

C

8 cm

Method 1. Relabel

A

A

B 6cm b C

c

5 cm

8 cm a

B

2. Which way up

sinA sinB sinC   a b c

2. Which form

We want angle C, so use c2 = a2 + b2 – 2ab cosC

sinA sinB sin120 o   a 5 6

3. Substitute in

52 = 82 + 62 – 2(8)(6)cosC

3. Substitute in

4. Rearrange to find cosC

25 = 64 + 36 – 96 cosC -75 = -96cosC cos C = -75  -96 = 0.78125

5. Use cos-1 to find C

C = 38.6o (3 SF)

4. Cross out

5. Rearrange to find sinB 6. Use sin-1 to find B

sinA sinB sin120 o   a 5 6 sin120 o  5 = 0.721… 6 B = 46.2o (3SF)

sinB =

Tip: As a check, make sure you have the smallest side opposite the smallest angle and the largest side opposite the largest angle.

3


   

Trigonometry 1 

4. Find the lettered angle or side in the following triangles

How do I know when to use which rule? You should only use the sine or cosine rule if the triangle is not rightangled.
You use the sine rule if you know a side together with the angle opposite it (eg A and a)
You use the cosine rule otherwise

70o

7 cm

a

40o

Questions 1. Find the lettered sides in the following triangles 6 cm

b

7 cm

2 cm

5 cm

40o

a

8 cm

C

8 cm

6 cm

50o

4 cm D

8 cm

b 2 cm

5. Force A is of magnitude 6N and acts vertically upwards. Force B is of magnitude 8N and acts at an angle of 30o to the vertical.

6 cm

a) Draw a triangle of forces to show the resultant of A and B. b) Calculate the magnitude and direction of the resultant of A and B 2. Find the lettered sides and angles in the following triangles. Answers (all given correct to 3SF ) 1. a = (62 – 22) = 32 = 5.66 cm b = (62 + 22) = 40 = 6.32 cm

a 1 cm

30o

10 cm

75o

2. a = 1  tan75o = 3.73 cm b = 3 cos40o = 3.92 cm C = tan-1(8 2) = 76.0o d = 10  sin30o = 5.00cm E = cos-1(3 4) = 41.4o F = 7 sin65o = 7.72 cm

d 3 cm 40o b

8 cm

E

4 cm

3 cm

7 cm

3. a) BD = 10  sin40o = 6.43 cm b) AB = (12 + 6.43..2) = 6.51cm

65

8 o  sin 40 = 5.47 cm sin 70 o b2 = 72 + 82 – 2(7)(8)cos400  b = 5.22 cm 62 = 72 + 82 – 2(7)(8)cosC  C = cos-1(0.6875) = 46.6o sin 50 o sinD =  5  D = sin-1(0.957…) = 73.2o 4

4. a =

o

f

C 2 cm

5.a) 3.

8N

B

150o 6N

10 cm

40o

A 1 cm

R

b) Magnitude: R2 = 82 + 62 – 2(8)(6)cos150o  R = 13.5N (3 SF) C Direction: sinx=

D

In the diagram above, find a) BD b) AB

sin150o  8  x = sin-1(0.295…)= 17.2o (3SF) 13.5...

Acknowledgements: This Factsheet was researched and written by Cath Brown Curriculum Press, Unit 305B The Big Peg, 120 Vyse Street, Birmingham B18 6NF. Physics Factsheets may be copied free of charge by teaching staff or students, provided that their school is a registered subscriber. They may be networked for use within the school. No part of these Factsheets may be reproduced, stored in a retrieval system or transmitted in any other form or by any other means without the prior permission of the publisher. ISSN 1351-5136

4