TRANSFORMATION GEOMETRY (1)
Learning Outcomes and Assessment Standards
Learning Outcome 3: Space, shape and measurement
Assessment Standard 12.3.4 (a)
• Use the compound angle identities to generalize the effect on the coordinates of the point (x; y)
after rotation about the origin through any angle θ.
Lesson
35
Overview
In this lesson you will:
●
determine the coordinates of the image of a point if this point is rotated about
the origin by any given angle θ.
Lesson
Rotations through any angle about the origin
DVD
In the following diagram, the angle of inclination of the line segment OP is given
by the angle θ. The point P(x; y) is rotated about the origin through an angle α and
its image is the point P(x; y). Let the radius of the circle be equal to 1.
Using trigonometry, we can rewrite the coordinates of point P as follows:
cos θ = _1x
∴ cos θ = x
∴ x = cos θ
y
sin θ = _1
∴ sin θ = y
∴ y = sin θ
Therefore the point P can be written as P(cos θ; sin θ)
Using trigonometry, we can rewrite the coordinates of point P as follows:
cos (θ + α) = _x1
∴ cos (θ + α) = x
∴ x = cos (θ + α)
y
sin (θ + α) = _1
∴ sin (θ + α) = y
∴ y = sin (θ + α)
If we now use compound angles and expand the expressions for x and y, the
following results emerge:
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1
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x = cos (θ + α)
∴ x = cos θ cos α – sin θ sin α
∴ x = x cos α – y sin α
y = sin (θ + α)
∴ y = sin θ cos α + cos θ sin α
∴ y = y cos α + x sin α
Therefore the point P can be written as P(x cos α – y sin α; y cos α + x sin α).
Therefore, in general, the coordinates of the image P of any point after rotation
about the origin through an angle of α is given by:
P(x cos α – y sin α; y cos α + x sin α)
Example 1
Determine the image of the point P(1; 3) after an anti-clockwise rotation about the
origin, through an angle of 120°.
x = x cos α – y sin α
∴ x = 1cos 120° – 3sin 120°
∴ x = cos (180°– 60°) – 3sin (180° – 60°)
∴ x = –cos 60° – 3sin 60°
_
√
∴ x = – _1 – 3 _3
( )
_ 2
2
–1 – 3√3
∴ x = _
2
y = ycos α + xsin α
∴ y = 3cos 120° + 1sin 120°
∴ y = 3cos (180° – 60°) + sin (180° – 60°)
∴ y = 3(–cos 60°) + sin 60°
_
√
∴ y = 3(– _1 ) + _3
2_
–3 + 3
∴ y = _
2
2
√
2
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(
_
_
)
√
–1 – 3√3 _
Therefore the image of P is the point P _
; –3 +2 3
2
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Activity 1
1. If the point A(1; –4) is rotated anti-clockwise about the origin through an
angle of 45°, determine the coordinates of A, the image of A.
2. If the point C(–3; 1) is rotated anti-clockwise about the origin through an
angle of 150°, determine the coordinates of C, the image of C.
Example 2
_
Determine the image of the pointP(–2; √3 ) after a clockwise rotation about the
origin, through an angle of 210°.
x = xcos α – ysin α
_
∴ x = (–2)cos (–210°) – (√3 )sin (–210°)
_
∴ x = (–2)cos 210° – (√3 )(–sin 210°)
_
∴ x = (–2)cos (180° + 30°) + (√3 )sin (180° + 30°)
_
∴ x = (–2)cos (–cos 30°) + (√3 )(–sin 30°)
_
∴ x = 2cos 30°– √3 sin 30°
_
_
√
∴ x = 2 _3 – √3 (_1)
(_2 ) _
2
2√3 – √3
∴ x = _
_ 2
√
∴ x = _3
2
y = ycos α + xsin α
_
∴ y = (√3 )cos (–210°) + (–2)sin (–210°)
_
∴ y = (√3 )cos 210° + (–2)(–sin 210°)
_
∴ y = (√3 )cos (180° + 30°) + 2sin (180° + 30°)
_
∴ y = (√3 )(–cos 30°) + 2(–sin 30°)
_ √_
∴ y = (√3 ) – _3 – 2(_1 )
( 2)
–3 – 2
= – _5
∴ y = _
2
2
2
(
_
)
√3
; – _52
Therefore the image of P is the point P _
2
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Activity 2
1. If the point B(–1; 4) is rotated clockwise about the origin through an angle of
45°, determine the coordinates of B, the image of B.
2. If the point D(1; –4) is rotated clockwise about the origin through an angle of
150°, determine the coordinates of D, the image of D.
Revision of compound angles
Before you attempt Activity 3, it is necessary to revise compound angles.
Example 3
Determine the value of sin 75° in surd form.
sin 75°
= sin (45° + 30°)
= sin 45°cos 30° + cos 45°sin 30°
_
_
_
√
√
√
= _2 _3 + _2 (_1 )
( _2 )( 2_ ) ( 2 ) 2
6+ 2
=_
4
√
√
The angle 75° can be seen as a rotation of 45° followed by a rotation of 30°.
You will use these ideas in Activity 3.
Activity 3
1. A point E(1; 2) is rotated about the origin through an angle of 30° to E and
then through an angle of 120° to E. Determine the coordinates of E and E by
using each rotation separately and then by using a single rotation.
2. (a) Determine the value of the following in surd form:
(i) sin 75°
(ii) cos 75°
(b) Hence determine the coordinates of F, the image of point F(–1; 2) if the
point F is rotated anti-clockwise about the origin through an angle of 75°.
4
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