A student converts 90° to radians and writes the answer as 1.5708. What is the problem with this answer?
AThe conversion is wrong — 90° does not equal π/2 radians
BThe student used the wrong conversion factor (180/π instead of π/180)
CRadian answers should be expressed as exact multiples of π (π/2), not as decimal approximations
DRadians cannot be compared to degrees without specifying the radius of the circle
1.5708 ≈ π/2, so the numerical value is correct — but the form is wrong. Radian answers should almost always be left as exact multiples of π. Writing 1.5708 instead of π/2 introduces rounding error, makes further calculations messier, and obscures the structure of the angle. Exact form is preferable for the same reason you write 1/3 rather than 0.333... — it's cleaner, more precise, and reveals the underlying relationship.
Question 2 Multiple Choice
If you forget the conversion formula, how can you re-derive the factor for converting degrees to radians?
ARecall that 1 radian ≈ 57.3° and use that approximation
BFrom the fundamental equivalence 360° = 2π radians, divide both sides by 360 to get 1° = π/180 radians
CUse the unit circle: set the radius to 1 and measure arc length in degrees
DThe formula must be memorized — there is no way to derive it from first principles
The single foundational fact is 360° = 2π radians — a full revolution in both measurement systems. Dividing both sides by 360 gives 1° = π/180 rad, so to convert degrees to radians, multiply by π/180. To go the other direction, divide both sides by 2π to get 1 rad = 180/π degrees. This derivation takes ten seconds and means you never need to memorize the formula outright.
Question 3 True / False
The derivative of sin(x) equals cos(x) only when x is measured in radians, not when x is in degrees.
TTrue
FFalse
Answer: True
This is one of the most important practical consequences of radian measure. The calculus identity d/dx[sin(x)] = cos(x) relies on the limit (sin h)/h → 1 as h → 0, which holds only when h is in radians. If x is in degrees, the formula gains a factor of π/180. This is why every trigonometric formula in calculus and beyond assumes radian input — and why fluency in radian conversion is prerequisite knowledge for calculus.
Question 4 True / False
Since π ≈ 3.14, an angle of π radians is approximately equal to 3.14 degrees.
TTrue
FFalse
Answer: False
This is a category error. The number π ≈ 3.14 is just that — a number. In radian measure, π radians means 'π times 1 radian,' which equals 180 degrees (half a full turn). The decimal 3.14 tells you the numerical value of π, not its degree equivalent. Saying 'π radians ≈ 3.14 degrees' confuses the number π with the angle measurement. This misconception often trips students who conflate the symbol π with the angle, rather than recognizing π as a coefficient in the expression.
Question 5 Short Answer
Explain why radian answers should be left as exact multiples of π rather than converted to decimal approximations.
Think about your answer, then reveal below.
Model answer: Exact multiples of π preserve the precise relationship between the angle and the underlying geometry. π/4 immediately tells you this is one-eighth of a full turn; 0.785 does not. Exact form avoids accumulated rounding error in subsequent calculations, keeps algebraic manipulation clean (π/4 × 2 = π/2, whereas 0.785 × 2 = 1.570 is already imprecise), and is required for the calculus formulas that use radian measure.
The deeper point is that 'leaving π in the answer' is not laziness — it is precision. The decimal expansion of π is infinite and non-repeating, so any decimal approximation loses information. In the same way that you write √2 instead of 1.414, you write π/3 instead of 1.047. This habit becomes especially important in calculus, where exact radian expressions simplify differentiation and integration of trigonometric functions.