1 of 39

P1 Chapter 6 :: Trigonometric Ratios

2 of 39

Chapter Overview

1:: Sine/Cosine Rule

3:: Graphs of Sine/Cosine/Tangent

2:: Areas of Triangles

There is technically no new content in this chapter since GCSE.

However, the problems might be more involved than at GCSE level.

3 of 39

RECAP :: Right-Angled Trigonometry

hyp

adj

opp

Remember that a ratio just means the ‘relative size’ between quantities (in this case lengths). For this reason, sin/cos/tan are known as “trigonometric ratios”.

?

4

?

4 of 39

Just for your interest…

Have you ever wondered why “cosine” contains the word “sine”?

Therefore these angles are complementary.

i.e. The cosine of an angle is the sine of the complementary angle.

Hence cosine = COMPLEMENTARY SINE

?

ME-WOW!

5 of 39

OVERVIEW: Finding missing sides and angles

You have	You want	Use
#1: Two angle-side opposite pairs	Missing angle or side in one pair	Sine rule
#2 Two sides known and a missing side opposite a known angle	Remaining side	Cosine rule
#3 All three sides	An angle	Cosine rule
#4 Two sides known and a missing side not opposite known angle	Remaining side	Sine rule twice

?

When triangles are not right-angled, we can no longer use simple trigonometric ratios, and must use the cosine and sine rules.

6 of 39

Cosine Rule

We use the cosine rule whenever we have three sides (and an angle) involved.

How are sides labelled ?

Calculation?

Proof at end of PowerPoint.

7 of 39

Dealing with Missing Angles

Label sides then substitute into formula.

Simplify each bit of formula.

Rearrange (I use ‘subtraction swapsie trick’ to swap thing you’re subtracting and result)

?

You have	You want	Use
All three sides	An angle	Cosine rule

8 of 39

Harder Ones

?

9 of 39

Test Your Understanding

?

1

3

?

2

?

Fro Note: You will get an obtuse angle whenever you inverse cos a negative value.

10 of 39

Exercise 6A

Pearson Pure Mathematics Text Book

Pages 108-110

1

2

?

11 of 39

Solutions to Extension Question 2

?

12 of 39

The Sine Rule

65°

85°

30°

10

5.02

9.10

For this triangle, try calculating each side divided by the sin of its opposite angle. What do you notice in all three cases?

c

C

b

B

a

A

?

You have	You want	Use
#1: Two angle-side opposite pairs	Missing angle or side in one pair	Sine rule

13 of 39

Examples

45°

8

11.27

85°

?

Q1

You have	You want	Use
#1: Two angle-side opposite pairs	Missing angle or side in one pair	Sine rule

100°

8

15.76

30°

?

Q2

50°

14 of 39

Examples

85°

6

5

56.11°

?

Q3

8

40.33°

10

126°

?

Q4

15 of 39

Exercise 6B

Pearson Pure Mathematics Year 1/AS

Pages 112=-114

1

?

16 of 39

Test Your Understanding

10

5

?

17 of 39

Exercise 6C

Pearson Pure Mathematics

Pages 115-116

18 of 39

Area of Non Right-Angled Triangles

59°

3cm

7cm

Area = 0.5 x 3 x 7 x sin(59)

= 9.00cm²

🖉

?

Fro Tip: You shouldn’t have to label sides/angles before using the formula. Just remember that the angle is between the two sides.

19 of 39

Test Your Understanding

?

20 of 39

Exercise 6D

Pearson Pure Mathematics Year 1/AS

Pages 117-118

21 of 39

Sin or cosine rule?

Sine

Recall that whenever we have two “side-angle pairs” involved, use sine rule. If there’s 3 sides involved, we can use cosine rule. Sine rule is generally easier to use than cosine rule.

Cosine

Sine

Cosine

Sine

Cosine

Sine

22 of 39

Using sine rule twice

You have	You want	Use
#4 Two sides known and a missing side not opposite known angle	Remaining side	Sine rule twice

?

23 of 39

Using sine rule twice

You have	You want	Use
#4 Two sides known and a missing side not opposite known angle	Remaining side	Sine rule twice

1: We could use the sine rule to find this angle.

2: Which means we would then know this angle.

🖉

?

24 of 39

Test Your Understanding

?

25 of 39

Problem Solving With Sine/Cosine Rule

a

b

c

?

26 of 39

Exercise 6E

Pearson Pure Mathematics Text Book

Pages 120-122

Solutions to extension problems on next slides.

1

2

27 of 39

Solution to Extension Problem 1

28 of 39

Solution to Extension Problem 2

This is FM content, but see a few lines below.

29 of 39

Sin Graph

What does it look like?

90

180

270

360

-90

-180

-270

-360

?

30 of 39

Sin Graph

What do the following graphs look like?

90

180

270

360

-90

-180

-270

-360

Suppose we know that sin(30) = 0.5. By thinking about symmetry in the graph, how could we work out:

sin(150) = 0.5

sin(-30) = -0.5

sin(210) = -0.5

?

31 of 39

Cos Graph

What do the following graphs look like?

90

180

270

360

-90

-180

-270

-360

?

32 of 39

Cos Graph

What does it look like?

90

180

270

360

-90

-180

-270

-360

Suppose we know that cos(60) = 0.5. By thinking about symmetry in the graph, how could we work out:

cos(120) = -0.5

cos(-60) = 0.5

cos(240) = -0.5

?

33 of 39

Tan Graph

What does it look like?

90

180

270

360

-90

-180

-270

-360

?

34 of 39

Tan Graph

What does it look like?

90

180

270

360

-90

-180

-270

-360

?

35 of 39

Transforming Trigonometric Graphs

There is no new theory here: just use your knowledge of transforming graphs, i.e. whether the transformation occurs ‘inside’ the function (i.e. input modified) or ‘outside’ the function (i.e. output modified).

?

36 of 39

Transforming Trigonometric Graphs

?

37 of 39

Exercise 6F/6G

Pearson Pure Mathematics

Pages 125, 128-129

?

1

Extension

2

3

38 of 39

APPENDIX :: Proof of Cosine Rule

39 of 39

APPENDIX :: Proof of Sine Rule