Prerequisites: lines, exponents, logs, trig

Trigonometric functions and radians

Radians measure an angle by arc length on a unit circle: 180∘=π180^\circ=\pi radians.

180∘=π rad,ℓ=rθ\begin{gathered} 180^\circ=\pi\ \text{rad}, \\[4pt] \ell=r\theta \end{gathered}

Radians measure an angle by the arc length it cuts on a circle of radius 1. Since 180∘=π180^\circ=\pi radians, multiply by π180\dfrac{\pi}{180} to go from degrees to radians and by 180π\dfrac{180}{\pi} to go back.

π6=30∘,π4=45∘,π2=90∘\begin{gathered} \frac{\pi}{6}=30^\circ, \\[4pt] \frac{\pi}{4}=45^\circ, \\[4pt] \frac{\pi}{2}=90^\circ \end{gathered}

An arc of radius rr and central angle θ\theta (in radians) has length rθr\theta. The rule (sin⁡x)′=cos⁡x(\sin x)'=\cos x holds only in radians, which is why calculus always uses them.

xyθarc length = θr = 1cos θsin θ
On a circle of radius 1, the angle θ in radians is the arc length itself, and the point is (cos θ, sin θ). A full turn is 2π.Drag the dot

Examples

  1. 01How many degrees?3π2\displaystyle \frac{3\pi}{2}AnswerHide
    Answer270∘\displaystyle 270^\circ

    Steps

    1. π\displaystyle \pi=180∘\displaystyle = 180^{\circ}
    2. Using this
    3. 3π2×180π\displaystyle \frac{3\pi}{2} \times \frac{180}{\pi}=270\displaystyle = 270
  2. 02What is the value?sin⁡π2\displaystyle \sin \frac{\pi}{2}AnswerHide
    Answer1\displaystyle 1

    Steps

    1. sin⁡π2\displaystyle \sin \frac{\pi}{2}=1\displaystyle = 1cos⁡π2\displaystyle \cos \frac{\pi}{2}=0\displaystyle = 0(1)2+(0)2\displaystyle (1)^2 + (0)^2=1\displaystyle = 1
  3. 03Arc length for radius rr and central angle θ\theta (radians)?\displaystyle Ar2θ\displaystyle r^2\thetaBθr\displaystyle \frac{\theta}{r}Crθ2\displaystyle \frac{r\theta}{2}Drθ\displaystyle r\thetaAnswerHide
    Answerrθ\displaystyle r\theta

    Steps

    1. θ=2π\displaystyle \theta = 2\pi⇒rθ=2πr\displaystyle \Rightarrow r\theta = 2\pi r
  4. 04How many radians?30∘\displaystyle 30^\circAnswerHide
    Answerπ6\displaystyle \frac{\pi}{6}

    Steps

    1. 180∘\displaystyle 180^{\circ}=π\displaystyle = \pi
    2. Using this
    3. 30×π180\displaystyle 30 \times \frac{\pi}{180}=π6\displaystyle = \frac{\pi}{6}
  5. 05What is the value?cos⁡π\displaystyle \cos \piAnswerHide
    Answer−1\displaystyle -1

    Steps

    1. sin⁡π\displaystyle \sin \pi=0\displaystyle = 0cos⁡π\displaystyle \cos \pi=−1\displaystyle = -1(0)2+(−1)2\displaystyle (0)^2 + (-1)^2=1\displaystyle = 1
  6. 06How many degrees?π6\displaystyle \frac{\pi}{6}AnswerHide
    Answer30∘\displaystyle 30^\circ

    Steps

    1. π\displaystyle \pi=180∘\displaystyle = 180^{\circ}
    2. Using this
    3. π6×180π\displaystyle \frac{\pi}{6} \times \frac{180}{\pi}=30\displaystyle = 30

FAQ

Why does calculus use radians?
Because lim⁡x→0sin⁡xx=1\lim_{x\to 0}\dfrac{\sin x}{x}=1 holds only in radians. In degrees you would get (sin⁡x)′=π180cos⁡x(\sin x)'=\dfrac{\pi}{180}\cos x, with an extra constant.

Published 2026-09-04 · Updated 2026-10-14