Circular Functions
C H A P T E R
16
Circular Functions
Objectives
To use radians and degrees for the measurement of angle. To convert radians to degrees and vice versa. To define the circular functions sine, cosine and tangent. To explore the symmetry properties of circular functions. To find standard exact values of circular functions. To understand and sketch the graphs of circular functions.
16.1
Measuring angles in degrees and radians
The diagram shows a unit circle, i.e. a circle of radius 1 unit. The circumference of the unit circle = 2 × 1 = 2 units ∴ the distance in an anticlockwise direction around the circle from A to B = A to C = 2 units
y
1 B
C –1
0 –1 D y 1 P
A 1
x
units 3 units A to D = 2
Definition of a radian
In moving around the circle a distance of 1 unit from A to P, the angle POA is defined. The measure of this angle is 1 radian. One radian (written 1c ) is the angle subtended at the centre of the unit circle by an arc of length 1 unit.
–1
0 –1
1 unit 1c A 1
x
445
ISBN 978-1-107-67331-1 © Michael Evans et al. 2011 Photocopying is restricted under law and this material must not be transferred to another party. Cambridge University Press
446
Essential Mathematical Methods 1 & 2 CAS
Note: Angles formed by moving anticlockwise around the circumference of the unit
circle are defined as positive. Those formed by moving in a clockwise direction are said to be negative.
Degrees and radians
The angle, in radians, swept out in one revolution of a circle is 2 c .
∴ ∴ ∴
2
c c
= 360◦ = 180◦ 180◦ or 1◦ = c 1c =
180
Example 1 Convert 30◦ to radians. Solution 180 30 × 30◦ = 180 1◦ = c ∴
=
c
6
Example 2 c Convert
4
to degrees. 180◦ × 180 = 45◦ 4× 4 rather
Solution 1c =
∴
c
4
=
Note: Often the symbol for radian, c , is omitted. For example, angle 45◦ is written as c than
4
.
ISBN 978-1-107-67331-1 © Michael Evans et al. 2011
16
Circular Functions
Objectives
To use radians and degrees for the measurement of angle. To convert radians to degrees and vice versa. To define the circular functions sine, cosine and tangent. To explore the symmetry properties of circular functions. To find standard exact values of circular functions. To understand and sketch the graphs of circular functions.
16.1
Measuring angles in degrees and radians
The diagram shows a unit circle, i.e. a circle of radius 1 unit. The circumference of the unit circle = 2 × 1 = 2 units ∴ the distance in an anticlockwise direction around the circle from A to B = A to C = 2 units
y
1 B
C –1
0 –1 D y 1 P
A 1
x
units 3 units A to D = 2
Definition of a radian
In moving around the circle a distance of 1 unit from A to P, the angle POA is defined. The measure of this angle is 1 radian. One radian (written 1c ) is the angle subtended at the centre of the unit circle by an arc of length 1 unit.
–1
0 –1
1 unit 1c A 1
x
445
ISBN 978-1-107-67331-1 © Michael Evans et al. 2011 Photocopying is restricted under law and this material must not be transferred to another party. Cambridge University Press
446
Essential Mathematical Methods 1 & 2 CAS
Note: Angles formed by moving anticlockwise around the circumference of the unit
circle are defined as positive. Those formed by moving in a clockwise direction are said to be negative.
Degrees and radians
The angle, in radians, swept out in one revolution of a circle is 2 c .
∴ ∴ ∴
2
c c
= 360◦ = 180◦ 180◦ or 1◦ = c 1c =
180
Example 1 Convert 30◦ to radians. Solution 180 30 × 30◦ = 180 1◦ = c ∴
=
c
6
Example 2 c Convert
4
to degrees. 180◦ × 180 = 45◦ 4× 4 rather
Solution 1c =
∴
c
4
=
Note: Often the symbol for radian, c , is omitted. For example, angle 45◦ is written as c than
4
.
ISBN 978-1-107-67331-1 © Michael Evans et al. 2011