SME
Grade 11 Math · Problem type
Coterminal and reference angles
Use this for “Find the coterminal angle of −495° from 0° to 360°, its quadrant and its reference angle.”
Diagram
θ
=
−495°:
the
point
(cos
θ,
sin
θ)
is
(−
√
2
2
,
−
√
2
2
).
1 full turn and 135° more, clockwise: the same point as 225°.
Reference angle: 45°.
Coterminal angle from 0° to 360°: 225°
Quadrant: 3
Reference angle: 45°
This is an example. Type your own values to start a new problem.
θ
Angle
example · °
c
Coterminal angle from 0° to 360°
worked out (not typed) · °
Q
Quadrant
worked out (not typed)
R
Reference angle
worked out (not typed) · °
Formulas
c
=
θ
+
360k,
with
k
whole,
from
0°
up
to
360°
225 = −495 + 360k, with k whole, from 0° up to 360°
Q
=
⌊
c
90
⌋
+
1
3
=
⌊
225
90
⌋
+
1
R
=
reference
angle
of
c
in
quadrant
Q
45 = reference angle of 225 in quadrant 3
Assumptions
•
Adding or taking away whole turns of 360° ends on the same side: a coterminal angle.
•
The reference angle is the acute angle to the x-axis; the quadrant gives the signs.
•
An angle on an axis (a multiple of 90°) lies between quadrants: it has no quadrant or reference angle.
Step-by-step
WE KNOW
θ = −495°
FIND
coterminal angle from 0° to 360° (c), quadrant (Q), reference angle (R)
Step 1 · Find coterminal angle from 0° to 360° (c)
Use c = θ + 360k, with k whole, from 0° up to 360°
Add or take away whole turns of 360° until the angle is from 0° to 360°.
c
=
θ
+
2
×
360
c
=
−495
+
2
×
360
c
=
−495
+
720
c
=
225°
Step 2 · Find quadrant (Q)
Use Q = ⌊c ÷ 90⌋ + 1
225° is between 180° and 270°: quadrant III.
Q
=
⌊
225
90
⌋
+
1
Q
=
3
Step 3 · Find reference angle (R)
Use R = reference angle of c in quadrant Q
In quadrant III, measure past 180°.
R
=
c
−
180
R
=
225
−
180
R
=
45°
Check
Put the numbers back in. Both sides should match.
225 = −495 + 2 × 360 ✓
3
=
⌊
225
90
⌋
+
1
✓
45 = 225 − 180 ✓
Topic check
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