Degree Measurement of Angles
One full revolution makes an angle of $360^{\circ}$, and the angle on a straight line is $180^{\circ}$. Therefore, one degree, $1^{\circ}$, can be defined as $\dfrac{1}{360}$ of one full revolution.
For greater accuracy we define one minute, $1’$, as $\dfrac{1}{60}$ of one degree and one second, $1^{\prime \prime}$, as $\dfrac{1}{60}$ of one minute.
Radian Measurement of Angles
One new area is the concept of a radian. We have already been familiar with measuring angles in degrees ($^{\circ}$) and will recall that there are $360^{\circ}$ in a full circle. An alternative unit for angle measurement is the radian.
An angle is said to have a measure of $1$ radian ($^c$) if it is subtended at the circle’s centre by an equal length to the radius. The symbol ‘$c$’ is used for radian measure but is usually omitted. By contrast, the degree symbol is always used when the measure of an angle is given in degrees.
It can be seen that $1^c$ is slightly smaller than $60^{\circ}$.
$$1^c \approx 57.2958^{\circ}$$
Degree-Radian Conversions
$$\pi^c \equiv 180^{\circ}$$
To convert from degrees to radians, we multiply by $\dfrac{\pi}{180^{\circ}}$.
For example, $30^{\circ} \equiv 30^{\circ} \times \dfrac{\pi}{180^{\circ}} = \dfrac{\pi}{6}$
To convert from radians to degrees, we multiply by $\dfrac{180^{\circ}}{\pi}$.
For example, $\dfrac{\pi}{3} \equiv \dfrac{\pi}{3} \times \dfrac{180^{\circ}}{\pi} = 60^{\circ}$
$$\text{Degrees} \times \dfrac{\pi}{180^{\circ}} = \text{Radians} \\
\text{Radians} \times \dfrac{180^{\circ}}{\pi} = \text{Degrees}$$
Example 1
Convert $45^{\circ}$ to radians in terms of $\pi$.
\( \begin{align} \displaystyle
45^{\circ} &= 45^{\circ} \times \dfrac{\pi}{180^{\circ}} \\
&= \dfrac{\pi}{4}
\end{align} \)
Example 2
Convert $\dfrac{2 \pi}{3}$ radians to degrees.
\( \begin{align} \displaystyle
\dfrac{2 \pi}{3} &= \dfrac{2 \pi}{3} \times \dfrac{180^{\circ}}{\pi} \\
&= 120^{\circ}
\end{align} \)
Example 3
Convert $67^{\circ}$ to radians, correcting to three significant figures.
\( \begin{align} \displaystyle
67^{\circ} &= 67^{\circ} \times \dfrac{\pi}{180^{\circ}} \\
&= 1.16937 \cdots \\
&= 1.17
\end{align} \)
Example 4
Convert $1.2$ radians to degrees, correcting to three significant figures.
\( \begin{align} \displaystyle
1.2 &= 1.2 \times \dfrac{180^{\circ}}{\pi} \\
&= 68.75493 \cdots^{\circ} \\
&= 68.8^{\circ}
\end{align} \)
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