Two moves turn any angle into one you already know the values for.
Find the reference angle
Find the reference angle for −150°.
Add a full turn to land in range: −150 + 360 = 210°, which sits in Quadrant III. There the reference angle is the angle minus 180: 210 − 180 = 30°. B is the coterminal angle — that is only the first step, not the answer.
1
A full turn puts you back where you started.
So add or subtract 360° as often as it takes to land inside one revolution.
405 − 360 = 45°
one turn too many
−150 + 360 = 210°
negative means clockwise
positive angle
counter-clockwise
from the positive x-axis
negative angle
clockwise
same starting line
Coterminal means “same finishing line”
−150°, 210° and 570° all stop on the same ray, so every trig value they have is
identical. In radians the full turn is 2π — subtract 2π instead, and 9π4 becomes
π4.
2
Name the quadrant.
Quadrant
Angles
Ours
I
0° to 90°
45°
II
90° to 180°
—
III
180° to 270°
210°
IV
270° to 360°
—
Do this on the coterminal angle, not the original. −150° looks like it belongs
nowhere until you add the turn.
3
The reference angle is measured to the x-axis.
Always the x-axis. Never the y-axis, and never more than 90°.
Quadrant
Reference angle
Example
I
θ itself
45° → 45°
II
180 − θ
150° → 30°
III
θ − 180
210° → 30°
IV
360 − θ
300° → 60°
Every rule is just “how far from the horizontal”. If you forget which one to use,
sketch the ray and read the gap to the nearest half of the x-axis — it can only be
the answer.
4
The quadrant supplies the sign.
All Students Take Chemistry — reading anticlockwise from quadrant I.
Quadrant
Positive
sin
cos
tan
I — All
everything
+
+
+
II — Students
sine
+
−
−
III — Take
tangent
−
−
+
IV — Chemistry
cosine
−
+
−
A reciprocal keeps its partner's sign: csc matches sin, sec matches cos, cot
matches tan.
5
Put the two together.
Value from the reference angle, sign from the quadrant.
It is an acute angle, so it carries no sign at all. The minus in −½ came from the
quadrant, applied at the end — not from the −150°.
Angles on an axis (0°, 90°, 180°, 270°) have no quadrant and no reference angle. Read
them from the point where the ray meets the unit circle, with cos = x and sin = y:
90° is (0, 1), 180° is (−1, 0), 270° is (0, −1). So cos 180° = −1, sin 270° = −1, and
tan 90° does not exist (it would divide by x = 0).
6
Variations: radians past a full turn, two turns in quadrant II, a ratio with its quadrant.
Evaluate sin(11π3).
11π3 − 2π = 11π3 − 6π3 = 5π3
one turn is 2π = 6π3
5π3 = 300°, quadrant IV
reference 2π − 5π3 = π3
sin is negative in IV
only cosine is positive
= −sin(π3) = −√32
value, then sign
Different: a radian angle. Write 2π over the same denominator before subtracting.
Evaluate cos 855°.
855 − 720 = 135°
two full turns
quadrant II, reference 180 − 135 = 45°
the gap to the x-axis
= −cos 45° = −√22
cosine is negative in II
Different: quadrant II uses 180 − θ, and the angle needs more than one turn removed.
sin θ = 513 and θ is in quadrant II. Find cos θ and tan θ.
opposite 5, hypotenuse 13 → adjacent 12
5-12-13, sizes only
cos θ = −1213
x is negative in II
tan θ = 5−12 = −512
sin ÷ cos
The triangle gives sizes, the quadrant gives signs
The reference triangle only ever has positive sides, so it yields 1213 and 512.
Forgetting to attach quadrant II’s signs at the end gives the positive pair — both
wrong.
On the exam
Expect a big or negative angle to evaluate, often two of them added, or “sin is
negative and tan is positive — which quadrant?”. Bring the angle into 0°–360°, name
the quadrant, take the angle to the x-axis, and attach the ASTC sign last. Ten-second
check: sketch the ray — left means cos is negative, down means sin is negative.
✓
Check: does the sign match the picture?
In quadrant III the ray points left and down, so both coordinates are negative.
210° rayx < 0, y < 0left and down
sin = y−½negative ✓
tan = yxnegative ÷ negativepositive, as ASTC says ✓
Remember the flow
Add or subtract 360° · Name the quadrant · Angle to the x-axis · Apply the sign
Four steps, and every angle becomes 30, 45 or 60.
Now you try
Practice problem
Evaluate
Evaluate tan 240°.
240° is in Quadrant III with reference angle 240 − 180 = 60°, so the size is tan 60° = √3. In Q3 sine and cosine are both negative, so their ratio is positive: tan 240° = +√3.A takes the sign from sine alone.
