Questions: Radian Measure

Question 1

A student calculates the derivative of f(x) = sin(x°) — sine where x is measured in degrees — and correctly gets f'(x) = (π/180)cos(x°). Why does the extra factor π/180 appear?

• A. Because the derivative of sin is cos(x) only when x is in radians; rewriting sin(x°) as sin(πx/180) introduces the factor π/180 via the chain rule
• B. Because the student forgot to apply the power rule to x
• C. Because cosine has a different scale than sine when measured in degrees
• D. Because π/180 is a correction factor for the amplitude, not the angle

Answer: (A)

To differentiate sin(x°), rewrite it as sin(πx/180) — converting the degree argument to radians. The chain rule then gives: d/dx[sin(πx/180)] = cos(πx/180) · (π/180). This factor is unavoidable whenever angles are in degrees. The formula d/dx[sin(x)] = cos(x) is clean only when x is in radians, because radians are dimensionless ratios and the derivative of arc length with respect to radian angle is exactly 1.

Question 2

Why are radians considered the mathematically natural unit for angles?

• A. Because radians are dimensionless ratios (arc length divided by radius), which eliminates conversion factors in calculus formulas
• B. Because radians produce simpler numbers: π is smaller than 360
• C. Because international mathematics standards adopted radians in the 20th century for consistency
• D. Because radian values are easier to measure physically with a protractor

Answer: (A)

A radian is defined as θ = s/r — the ratio of arc length to radius. This ratio has no units (m/m = 1), making radians dimensionless. This is not aesthetic; it is functional. The derivative of sin(x) equals cos(x) exactly because radian measure makes the geometric limit lim(h→0) sin(h)/h = 1 hold without any correction factor. Degrees are an arbitrary historical division of the circle (360 chosen for astronomical reasons), and using them in calculus requires constant correction by π/180.

Question 3

A radian is a unit with a physical dimension, just as a meter or a second has a dimension.

• T. True
• F. False

Answer: False

Radians are dimensionless. A radian is defined as θ = s/r — the ratio of arc length to radius. Since both s and r have units of length, the ratio is unitless: m/m = 1. This is why radians can be 'cancelled' in calculations in a way that degrees cannot. When you write sin(π/2) = 1, the π/2 is just a number with no units. This dimensionlessness is precisely what makes radian measure mesh cleanly with calculus.

Question 4

The formula for arc length, s = rθ, is only valid when θ is measured in radians.

• T. True
• F. False

Answer: True

The arc length formula s = rθ follows directly from the definition of a radian: one radian is the angle for which s = r, so for a general angle θ radians, s = rθ. If θ is in degrees, the formula becomes s = rθ · (π/180), requiring a conversion factor. The clean form s = rθ works only in radians. Similarly, the sector area formula A = ½r²θ requires radian measure to hold without a conversion factor.

Question 5

Explain why the formula d/dx[sin(x)] = cos(x) only holds when x is measured in radians. What specifically goes wrong if x is in degrees?

Model answer: The derivative formula arises from the limit lim(h→0) sin(h)/h = 1. This limit equals 1 only when h is measured in radians. In radian measure, sin(h) ≈ h for small h (because arc length ≈ chord length when both equal radius × angle). If h is in degrees, sin(h°) ≈ (π/180)h for small h, so the limit becomes π/180 instead of 1. Consequently, d/dx[sin(x°)] = (π/180)cos(x°). The factor π/180 pollutes every trig derivative and integral when degrees are used.

The root cause is the definition: a radian makes the ratio of arc length to radius equal 1, so for small θ in radians, sin(θ) ≈ θ exactly (to first order). This is the geometric fact underlying the limit. In degrees, sin(1°) ≈ 0.01745 ≈ π/180, not 1 — the angle in degrees is 57.3 times larger than the corresponding value in radians, introducing a proportional correction. Radian measure is not merely a convention; it is the unit that makes the geometry consistent with the calculus.