12.2 Unit Circle, Radians, and Arc Length

12.2 Unit Circle, Radians, and Arc Length

College Board’s Table 11 trigonometry cluster includes arc length and radian measures alongside right-triangle trig. College Board’s Skills Insight™ statements for AAF place trigonometric functions — sine, cosine, tangent — including the unit circle in the top score band, 276–300. The 250–262 right-triangle work from Right-Triangle Trigonometry and Special Triangles still applies: sine is still opposite over hypotenuse. The unit circle is that same ratio with hypotenuse 1. Work FREE mixed AAF items at /practice/accuplacer-advanced-algebra. Official category names live on College Board’s What’s on the Tests page.

Degrees and radians

A radian is the central angle that cuts an arc equal in length to the radius. One full turn is 360° and also 2π radians, so the conversion identity is:

π radians = 180°

To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.

Worked: 240° in radians is 240 · π/180 = 4π/3. Worked the other way: 5π/6 in degrees is (5π/6) · (180/π) = 150°. Cancel π first so you are not tempted to punch a calculator and then round the exact option away.

AAF will mix the two units in one stem. If an equation is written sin θ = 1/2 for θ in [0, 2π), the answers are radians. If the interval is [0°, 360°), the answers are degrees. Do not convert unless the options force you to.

Coterminal angles

Two angles are coterminal when they share a terminal ray. Add or subtract any integer number of full turns:

θ + 360° · k in degrees, or θ + 2π · k in radians, for integer k.

Worked: 390° is coterminal with 30° because 390 − 360 = 30. So the unit-circle point at 390° is the same as the point at 30°: (√3/2, 1/2). Worked in radians: −π/6 is coterminal with 11π/6 because −π/6 + 2π = 11π/6. The point is (√3/2, −1/2) in Quadrant IV.

Reducing to a coterminal angle in [0, 2π) is the first move on many 276–300 items. AAF does not care that −π/6 is a negative rotation; it cares that you can name cos(−π/6) = √3/2.

The unit circle: (cos θ, sin θ)

The unit circle is the circle of radius 1 centered at the origin, equation x^2 + y^2 = 1. An angle θ in standard position has its vertex at the origin and its initial ray along the positive x-axis. The terminal ray meets the circle at a point P(x, y). By the right-triangle definition with hypotenuse 1:

x = cos θ

y = sin θ

That is the entire upgrade from Skills Insight 250–262 to 276–300. Adjacent over hypotenuse becomes the x-coordinate; opposite over hypotenuse becomes the y-coordinate. Tangent is still y/x whenever x ≠ 0, which is why tan θ blows up at odd multiples of π/2.

Common first-quadrant points

Drop a perpendicular from P to the x-axis and you recover a 30-60-90 or 45-45-90 triangle with hypotenuse 1.

Axis intercepts to memorize with those five: 180° = π is (−1, 0), 270° = 3π/2 is (0, −1), and 360° = 2π returns to (1, 0).

Reference angles and the other quadrants

A reference angle is the acute angle between the terminal ray and the x-axis. It copies the absolute values from the table above; the quadrant then attaches signs.

• Quadrant I (0° to 90°): reference = θ itself.
• Quadrant II (90° to 180°): reference = 180° − θ or π − θ.
• Quadrant III (180° to 270°): reference = θ − 180° or θ − π.
• Quadrant IV (270° to 360°): reference = 360° − θ or 2π − θ.

Worked: 210° = 7π/6 is 180° + 30°, so the reference is 30° and the point is in Quadrant III. Absolute values from 30° are √3/2 and 1/2. In QIII both cosine and sine are negative, so the point is (−√3/2, −1/2).

Worked: 2π/3 is 120°, reference 60°, Quadrant II. Absolute values from 60° are 1/2 and √3/2. In QII cosine is negative and sine is positive, so the point is (−1/2, √3/2).

Worked: 315° = 7π/4 is 360° − 45°, reference 45°, Quadrant IV. Point: (√2/2, −√2/2).

The same Pythagoras identity that gave you 5-12-13 now reads cos^2 θ + sin^2 θ = 1 on the circle. If a stem gives cos θ = −5/13 and θ in Quadrant II, then sin θ = +12/13 because sine is positive in QII and the 5-12-13 triple still supplies the unsigned 12.

Quadrant signs (ASTC)

A compact memory phrase is All Students Take Calculus, reading counterclockwise from Quadrant I:

• QI: All of sin, cos, and tan are positive.
• QII: Sine (and its reciprocal csc) is positive; cos and tan are negative.
• QIII: Tangent (and cot) is positive; sin and cos are both negative.
• QIV: Cosine (and sec) is positive; sin and tan are negative.

Tangent’s sign is sine’s sign divided by cosine’s sign, so QIII gets a positive tangent from a negative over a negative. Do not invent a fifth rule.

Arc length s = rθ

On a circle of radius r, a central angle θ measured in radians cuts an arc of length

s = rθ

That is the definition of radian measure rearranged: θ = s/r. If the stem gives degrees, convert first. Do not plug 150 into s = rθ — 150 is not a radian measure.

Worked in radians: Radius 9, central angle 2π/3. Then s = 9 · (2π/3) = 6π.

Worked from degrees: Radius 12, central angle 150°. Convert: 150° = 150 · π/180 = 5π/6. Then s = 12 · (5π/6) = 10π.

Worked for the angle: Radius 8, arc length 4π. Then θ = s/r = 4π/8 = π/2, which is 90° if the options are in degrees.

Worked for the radius: Arc 5π, angle π/4. Then r = s/θ = 5π / (π/4) = 20.

A full-circle check: θ = 2π gives s = 2πr, the circumference. A half-circle θ = π gives s = πr. If your formula spits out something larger than the circumference for an angle under 2π, you left the angle in degrees.

Related (not the same formula): a sector’s area is (1/2)r^2 θ with θ in radians. Table 11 names arc length, not sector area, but the same radian requirement applies. Geometry Algebra 1 already used 2πr as a circumference; this section is the fractional-turn version.

AAF traps for radians and the unit circle

• Arc length with degrees. s = rθ is illegal until θ is in radians. Convert with π/180.
• Swapping sine and cosine on the circle. The x-coordinate is cosine, including at 0 where cos 0 = 1 and sin 0 = 0. The point is (1, 0), not (0, 1).
• Wrong quadrant sign. 7π/6 is QIII, not QII. Both coordinates negative: (−√3/2, −1/2), not (−√3/2, 1/2).
• 30 versus 60 on the axes. At π/6 the y-coordinate is the smaller one 1/2. At π/3 the y-coordinate is the larger one √3/2. Mixing those two points is the highest-yield unit-circle miss.
• Coterminal leftover. −π/3 is not a Quadrant IV 60° reference until you add 2π and land on 5π/3, point (1/2, −√3/2).
• Using 180 in place of π. s = 12 · 150 / 180 without a π is a length of 10, not 10π. The π survives the conversion.

Handheld calculators stay restricted the same way as in 12.1. Exact points on the unit circle are faster than a decimal approximation of cos 210°. After you can name (cos θ, sin θ) at every 30°/45°/60° mark and compute s = rθ, you are ready for equivalent forms and graphs in Trigonometric Graphs and Identities.