\begin{definition}[Radian] A radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. \end{definition}
\begin{note} The radian is a unit of measurement for angles, just like degrees. \end{note}
\begin{example} The figure below shows a circle of radius \$r\$ with an arc of length \$r\$.
\begin{placeholder}[diagram]{A circle with radius r and an arc of length r.} \end{placeholder}
The angle subtended by the arc at the centre of the circle is \$1\$ radian. \end{example}
Radians and the Unit Circle
The unit circle is a circle with radius \$1\$.
\begin{note} The unit circle is a powerful tool in mathematics, especially in trigonometry and calculus. \end{note}
The circumference of a circle of radius \$r\$ is \$2\pi r\$.
Therefore, the circumference of the unit circle is \$2\pi\$.
\begin{example} The figure below shows the unit circle with its circumference divided into \$2\pi\$ equal parts.
\begin{placeholder}[diagram]{The unit circle with its circumference divided into 2π equal parts.} \end{placeholder}
Each part has length \$1\$ and subtends an angle of \$1\$ radian at the centre of the circle. \end{example}
Radians and Degrees
The degree is another unit of measurement for angles.
\begin{note} The degree is a more common unit of measurement for angles in everyday life. \end{note}
A full circle has \$360\$ degrees.
Therefore, \$2\pi\$ radians is equal to \$360\$ degrees.
\begin{example} The figure below shows the unit circle with its circumference divided into \$360\$ equal parts.
\begin{placeholder}[diagram]{The unit circle with its circumference divided into 360 equal parts.} \end{placeholder}
Each part has length \$\frac{1}{180}\$ and subtends an angle of \$1\$ degree at the centre of the circle. \end{example}
Converting Between Radians and Degrees
To convert an angle from radians to degrees, multiply by \$\frac{180}{\pi}\$.
To convert an angle from degrees to radians, multiply by \$\frac{\pi}{180}\$.
\begin{example}
Converting Radians to Degrees
Convert \$2\$ radians to degrees.
- Multiply by \$\frac{180}{\pi}\$:
$$\$\$2 \cdot \frac{180}{\pi} \approx 114.59\$\$$
- Therefore, \$2\$ radians is approximately \$114.59\$ degrees.
Converting Degrees to Radians
Convert \$90\$ degrees to radians.
- Multiply by \$\frac{\pi}{180}\$:
$$\$\$90 \cdot \frac{\pi}{180} = \frac{\pi}{2}\$\$$
- Therefore, \$90\$ degrees is equal to \$\frac{\pi}{2}\$ radians. \end{example}
\begin{self_review}
- Convert \$3\$ radians to degrees.
- Convert \$45\$ degrees to radians. \end{self_review}
\begin{tok} How do different units of measurement (e.g., radians and degrees) affect our understanding of mathematical concepts? \end{tok}
