Radians and values of sin and cos, with practice problems

Q: Why does calculus use radians?

Because \lim_{x\to 0}\frac{\sin x}{x}=1 holds only in radians. In degrees you would get (\sin x)'=\frac{\pi}{180}\cos x, with an extra constant.

\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 $180^\circ=\pi$ radians, multiply by $\frac{\pi}{180}$ to go from degrees to radians and by $\frac{180}{\pi}$ to go back.

\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 $r$ and central angle $\theta$ (in radians) has length $r\theta$ . The rule $(\sin x)'=\cos x$ holds only in radians, which is why calculus always uses them.

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

01How many degrees? $\displaystyle \frac{3\pi}{2}$ AnswerHide

Answer $\displaystyle 270^\circ$
02What is the value? $\displaystyle \sin \frac{\pi}{2}$ AnswerHide

Answer $\displaystyle 1$
03What is the value? $\displaystyle \cos \pi$ AnswerHide

Answer $\displaystyle -1$
04How many degrees? $\displaystyle \frac{\pi}{6}$ AnswerHide

Answer $\displaystyle 30^\circ$
05What is the value? $\displaystyle \cos \frac{\pi}{2}$ AnswerHide

Answer $\displaystyle 0$
06How many degrees? $\displaystyle \frac{3\pi}{4}$ AnswerHide

Answer $\displaystyle 135^\circ$