Cambridge International Mathematics

IMPORTANT TRIGONOMETRIC RATIOS IN THE UNIT CIRCLE

InChapter 15we found the trigonometric ratios for the angles 0 o, 30 o, 45 o, 60 oand 90 o.

μ cosμ sinμ tanμ

0 o 1 0 0

30 o

p

3

2

1

2

p^1

3

45 o p^12 p^121

60 o^12

p

3

2

p

3

90 o 0 1 undefined

These angles correspond

to the points shown on

the first quadrant of the

unit circle:

We can use the symmetry of the unit circle to find

the coordinates of all points with angles that are

multiples of 30 oand 45 o.

For example, the point Q corresponding to an angle

of 120 ois a reflection in they-axis of point P with

angle 60 o.

³

¡^12 ,

p 3

2

́

.

Multiples of 30 o Multiples of 45 o

We can find the trigonometric ratios of these angles using the coordinates of the corresponding point on the

unit circle.

Example 3 Self Tutor

Use a unit circle diagram to findsinμ,cosμandtanμfor:

a μ=60o b μ= 150o c μ= 225o

x

y

(0, 1)

(1, 0)

(0, 1)-

(,0)-1

³

1

2 ;

p 3

2

́

³p

3

2 ;

1

2

́

³p

3

2 ;¡

1

2

́

³

1

2 ;¡

p 3

2

³ ́

¡^12 ;¡

p

3

2

́

³

¡

p

3

2 ;¡

1

2

́

³

¡

p

3

2 ;

1

2

́

³

¡^12 ;

p 3

2

́

120°120°

210°210° OO

Q has the negative -coordinate and the same -coordinate as P, so the coordinates of Q arexy

x

y

(0, 1)

(1, 0)

(0, 1)-

(1,0)-

³

p^1

2 ;

p^1

2

́

³

p^1

2 ;¡

p^1

2

³ ́

¡p^12 ;¡p^12

́

³

¡p^12 ;p^12

́

45°

135°135°

225°

315°315°

OO

O

30°30°

45°45°

60°60°

y

x

³

1

2 ;

p

3

2

́

³

p^1

2 ;

p^1

2

́

³p

3

2 ;

1

2

́

()1 ¡0,

()0 ¡1,

-1 1

1

O

Q,()xy¡ P

y

x

60° 60°

120°

³

(^12) ;p 23
́
582 Further trigonometry (Chapter 29)
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Y:\HAESE\IGCSE01\IG01_29\582IGCSE01_29.CDR Monday, 27 October 2008 2:52:35 PM PETER