Measure an angle with the radius instead of a protractor, and the circle hands you its own coordinates.
What are the coordinates of P?
In the xy-plane, point P lies on the circle with center (0, 0) and radius 1. The angle
measured counterclockwise from the positive x-axis to ray OP has a measure of
4π3
radians.
4π3 is 240°, which lands in the third quadrant, 60° past the negative x-axis. On a unit circle the point is (cos θ, sin θ), and in quadrant III both are negative, so P = (−12,−√32). C uses a reference angle of 30° instead of 60° and so swaps the two numbers — it is the point at 210°. D keeps the correct pair but loses the sign on the x-coordinate, landing in quadrant IV. A has the right x but the wrong sign on y, which would be 120°, not 240°.
1
A radian is the angle whose arc is one radius long.
Bend the radius around the rim. The angle it opens up at the center is one radian — no protractor involved.
circumference = 2πr
the whole rim
2πr ÷ r = 2π
how many radii fit around it
2π radians = 360°
a full turn, said two ways
That is the whole reason 2π appears: it is how many radius-lengths it takes to go once
around any circle, large or small.
A radian is a length over a length, so it carries no unit symbol — an angle written without
a degree sign, such as 3π2, is already in radians. The same definition is what makes the
arc-length formula s = rθ work, and only with θ in radians.
2
π radians = 180°, and one factor converts both ways.
π rad = 180°
degrees → radians
× π180
the π goes on top
radians → degrees
× 180π
the π goes underneath
40 × π180 = 40π180
40° into radians
= 2π9
divide top and bottom by 20
5π3 × 180π = 5 × 60
5π3 into degrees; π cancels, 180 ÷ 3 = 60
= 300°
fourth quadrant
You do not have to remember which way up it goes
Both versions are π180 = 1, written so the unit you are leaving cancels: going to
radians the degrees must cancel, so 180 sits underneath; coming back, π does. Check the
size too — 40° is small, and 2π9 ≈ 0.70, well under the 1.57 of a quarter turn.
3
Read the denominator as a slice of 180°.
Sixths are 30° each, quarters are 45° each, thirds are 60° each. That single habit converts most SAT angles on sight.
Degrees
Radians
Where it lands
30°
π6
one sixth of a half turn
45°
π4
one quarter of a half turn
60°
π3
one third of a half turn
90°
π2
a quarter turn
120°
2π3
two 60° steps
135°
3π4
three 45° steps
180°
π
a half turn
270°
3π2
three quarters
360°
2π
a full turn
So 4π3 is four 60° steps — 240°. Counting steps beats multiplying by 180π, and it tells
you where the angle lands at the same time.
Leave π where it is
2π9 is the answer, not 0.698. SAT choices for these items are printed with π in them, so
a decimal will not match anything on the page.
4
On a circle of radius 1, the point is (cos θ, sin θ).
Drop a perpendicular from the point to the x-axis and a right triangle appears, with the radius as its hypotenuse.
cos θ = adjhyp = x1 = x
the hypotenuse is the radius, 1
sin θ = opphyp = y1 = y
dividing by 1 changes nothing
P = (cos θ, sin θ)
the coordinates are the ratios
Cosine is the across, sine is the up — in that order, because (x, y) is in that order.
Two consequences worth banking. Since x² + y² = 1 here, cos²θ + sin²θ = 1 for every angle.
And since neither coordinate can exceed the radius, sin θ and cos θ always sit between −1
and 1 — a choice of 54 for a sine is not a near miss, it is impossible.
5
The whole circle, in both units at once.
Sixteen angles, each written as degrees above and radians below. The gold spoke is the opening problem's 4π3.
Walk round it counterclockwise from (1, 0) and both columns climb together: 30° with π6,
45° with π4, 60° with π3, 90° with π2. A negative measure just means walking the other
way, so −π3 lands where 5π3 does.
Desmos draws this for you
Set the angle mode to Radians in the wrench menu, then plot (cos(4π3),sin(4π3)).
Desmos puts a dot exactly where the gold spoke meets the circle and reports the
coordinates as −0.5 and −0.866 — the decimals for −12 and −√32. Use it whenever you are
unsure of a sign.
6
The exact values come from two triangles.
Every value at 30°, 45° and 60° is read off these, using nothing but opposite, adjacent and hypotenuse.
θ
radians
cos θ
sin θ
the point
0°
0
1
0
(1, 0)
30°
π6
√32
12
right and a little up
45°
π4
√22
√22
equal parts of each
60°
π3
12
√32
a little right and well up
90°
π2
0
1
(0, 1)
The cosine column runs 1, √32,√22,12, 0 and the sine column is the same list backwards.
That is the complementary-angle identity showing itself: sin(x°) = cos(90° − x°).
The half that catches people
sin 30° = 12, not √32. The smaller angle faces the shorter side, so it takes the smaller
value. Check it against the circle: at 30° the point is only slightly above the x-axis, so
its height — the sine — must be small.
7
The quadrant sets the signs; the reference angle sets the digits.
Quadrant
Angles
cos θ (the x)
sin θ (the y)
I
0 to π2
+
+
II
π2 to π
−
+
III
π to 3π2
−
−
IV
3π2 to 2π
+
−
There is nothing to memorise: the signs are just which way along each axis you have walked.
Left of the y-axis, x is negative. Below the x-axis, y is negative.
θ = 4π3 = 4 × 60° = 240°
the opening problem
240° − 180° = 60°
the reference angle, measured to the x-axis
cos 60° = 12, sin 60° = √32
the digits, from the table
quadrant III → both negative
240° is past 180° and before 270°
P = (−12,−√32)
signs applied to the digits
P = (−12,−√32)
The reference angle is measured to the x-axis, not the y-axis
Choice C in the opening problem, (−√32,−12), is what you get from a reference angle of
30° instead of 60° — the angle up to the y-axis rather than back to the x-axis.
That one slip swaps the two coordinates every time.
sin19π4 = ?
bigger than 2π — go round first
19π4 − 4π = 19π − 16π4 = 3π4
4π is two full turns, which change nothing
sin3π4 = sin 135° = √22
quadrant II, reference 45°, sine positive
Angles differing by a whole number of full turns land on the same point, so they share a
sine and a cosine. Subtract 2π until the angle sits between 0 and 2π, then read it off.
✓
Check: does the point sit where it should?
the angle4π3 = 240°four 60° steps ✓
the quadrant180° < 240° < 270°III, so x and y both negative ✓
the size|−12| < |−√32|240° is closer to straight down ✓
on the circle?(14) + (34) = 1x² + y² = 1 ✓
Remember the flow
Convert · Find the quadrant · Reference angle · Apply the signs
x² + y² = 1 will catch almost every wrong pair.
Now you try
Practice problem
What is the value of a + b?
In the xy-plane, an angle in standard position has a measure of
5π4
radians. Its terminal ray intersects the unit circle centered at the origin at the point
(a, b).
5π4 is five 45° steps, or 225° — quadrant III, with a reference angle of 45°. There both coordinates are −√22, so a + b = −√22 + (−√22) = −√2. B adds only one of them, or stops after finding a single coordinate. C assumes the two cancel, which happens at 3π4 and 7π4, where the signs differ — not here, where they match. D drops both signs and answers for π4 instead.
Hint: count the 45° steps to place the angle on the circle before you touch a value. Once you
know the quadrant, the two digits are the same ones you would use at 45°, and only the signs
are left to decide.
