SME
Grade 12 Math · Problem type
Two turns in a row
Use this for “Turn (4, 2) by 30° and then by 60°. Which single matrix does both?”
Diagram
Rotate 30° counterclockwise about (0, 0), then rotate 60° counterclockwise about (0, 0).
A(4, 2) → A′(2.4641, 3.7321) → A″(−2, 4).
Both moves keep lengths and angles, so the final image is congruent to the figure.
This is an example. Type your own values to start a new problem.
R(
β
°)R(
α
°)
=
R(
γ
°)
c
cos γ
worked out (not typed)
s
sin γ
worked out (not typed)
x
x of the point
example
y
y of the point
example
x″
x of the final image
calculated
y″
y of the final image
calculated
Formulas
γ
=
α
+
β
90 = 30 + 60
c
=
cos(
γ
)
0 = cos(90°)
s
=
sin(
γ
)
1 = sin(90°)
x
″
=
c
×
x
−
s
×
y
−2 = 0 × 4 − 1 × 2
y
″
=
s
×
x
+
c
×
y
4 = 1 × 4 + 0 × 2
Assumptions
•
Doing α and then β is the product R(β)R(α), the later move written on the left.
•
For two turns about the origin the product is one turn by γ = α + β.
•
For most other pairs of moves the order changes the answer.
Step-by-step
WE KNOW
α = 30°, β = 60°, x = 4, y = 2
FIND
both turns (γ), cos γ (c), sin γ (s), x of the final image (x″), y of the final image (y″)
Step 1 · Find both turns (γ)
Use γ = α + β
R(
β
)R(
α
)
=
R(
α
+
β
):
multiplying
the
two
matrices
adds
the
angles
(the
sum
formulas).
γ
=
α
°
+
β
°
γ
=
30°
+
60°
γ
=
90°
Step 2 · Find cos γ (c)
Use c = cos(γ)
The
first
column
of
R(
γ
)
is
where
(1,
0)
lands:
(cos
γ
,
sin
γ
).
c
=
cos(90°)
c
=
0
Step 3 · Find sin γ (s)
Use s = sin(γ)
The
same
column’
s
second
entry.
s
=
sin(90°)
s
=
1
→
the
single
matrix
R(
γ
)
=
[[0,
−1],
[1,
0]]
Step 4 · Find x of the final image (x″)
Use x″ = c × x − s × y
The
first
row
of
R(
γ
),
[cos
γ
,
−sin
γ
],
times
the
column
(
x
,
y
).
x
″
=
0
×
4
−
1
×
2
x
″
=
0
−
2
x
″
=
−2
Step 5 · Find y of the final image (y″)
Use y″ = s × x + c × y
The
second
row
of
R(
γ
),
[sin
γ
,
cos
γ
],
times
the
column
(
x
,
y
).
y
″
=
1
×
4
+
0
×
2
y
″
=
4
Check
Put the numbers back in. Both sides should match.
90 = 30 + 60 ✓
0 = cos(90°) ✓
1 = sin(90°) ✓
−2 = 0 × 4 − 1 × 2 ✓
4 = 1 × 4 + 0 × 2 ✓
Topic check
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