39: Trigonometric ratios 39: Trigonometric ratios of 3 special anglesof 3 special angles

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 1: AS Core Vol. 1: AS Core ModulesModules

3 Special Angles

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Module C2

3 Special Angles

We sometimes find it useful to remember the trig. ratios for the angles 6045,30 and

These are easy to find using triangles.

In order to use the basic trig. ratios we need right angled triangles which also contain the required angles.

3 Special Angles

Consider an equilateral triangle.

Divide the triangle into 2 equal right angled triangles.

60 60

60 Trig ratios don’t depend on the size of the triangle, so we can let the sides be any convenient length.

22

2

( You’ll see why 2 is useful in a minute ).

3 Special Angles

Consider an equilateral triangle.

Divide the triangle into 2 equal right angled triangles.

60 60

30Trig ratios don’t depend on the size of the triangle, so we can let the sides be any convenient length.

1

22

2

We now consider just one of the triangles.

( You’ll see why 2 is useful in a minute ).

3 Special Angles

Consider an equilateral triangle.

Divide the triangle into 2 equal right angled triangles.

60

30Trig ratios don’t depend on the size of the triangle, so we can let the sides be any convenient length.

1

2

We now consider just one of the triangles.

( You’ll see why 2 is useful in a minute ).

3 Special Angles

1

2

From the triangle, we can now write down the trig ratios for 6030 and

Pythagoras’ theorem gives the 3rd side.3

60sin2

3 60cos 60tan2

1 3

30sin 30cos 30tan2

32

1

3

1

312 22

( Choosing 2 for the original side means we don’t have a fraction for the 2nd side )

60

30

3 Special Angles

1

1

45

45

211 22

2

45cos45sin 45tan2

11

For we again need a right angled triangle.

45

By making the triangle isosceles, there are 2 angles each of .45We let the equal sides have length 1.Using Pythagoras’ theorem, the 3rd side is

From the triangle, we can now write down the trig ratios for 45

3 Special Angles

2

1 45cos45sin 45tan 1

SUMMARY

60sin2

3 60cos 60tan2

13

30sin 30cos 30tan2

32

1

3

1

The trig. ratios for are: 6045,30 and

3 Special Angles

3 Special Angles

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

3 Special Angles

Consider an equilateral triangle.

Divide the triangle into 2 equal right angled triangles.

60 60

60 Trig ratios don’t depend on the size of the triangle, so we can let the sides be any convenient length.

22

2

( You’ll see why 2 is useful in a minute ).

3 Special Angles

1

2

From the triangle, we can now write down the trig ratios for 6030 and

Pythagoras’ theorem gives the 3rd side.3

60sin2

3 60cos 60tan2

1 3

30sin 30cos 30tan2

32

1

3

1

60

312 22

30

( Choosing 2 for the original side means we don’t have a fraction for the 2nd side )

3 Special Angles

1

1

45

45

211 22

2

45cos45sin 45tan2

11

For we again need a right angled triangle.

45

By making the triangle isosceles, there are 2 angles each of .45We let the equal sides have length 1.Using Pythagoras’ theorem, the 3rd side is

From the triangle, we can now write down the trig ratios for 45

3 Special Angles

2

1 45cos45sin 45tan 1

SUMMARY

60sin2

3 60cos 60tan2

13

30sin 30cos 30tan2

32

1

3

1

The trig. ratios for are: 6045,30 and