- $\sin\left(\frac{\pi}{2}\right)=\sin(90^\circ)=1$
- $\cos(\pi)=\cos(180^\circ)=-1$
- $\tan\left(\frac{\pi}{4}\right)=\tan(45^\circ)=1$
- $\text{cosec}\left(\frac{3\pi}{4}\right)=\frac{1}{\sin(135^\circ)}=\frac{1}{\frac{\sqrt{2}}{2}}=\sqrt{2}$
- $\text{sec}\left(\frac{5\pi}{4}\right)=\frac{1}{\cos(225^\circ)}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}$
- $\text{cotan}\left((2n+1)\pi/2\right)=0$ for odd multiples of $\pi/2$ because this is where tan has asymptotes.
Radians, Arc Length and Area of a Sector
Radians
Radians provide an alternative measurement for angles. They are particularly useful in calculus and finding the length of an arc or the area of a sector of a circle. We define 1 radian as the angle subtended when we traverse the part of a circle’s circumference that has the same length as its radius. (What does subtend mean?)
It follows that the angle required to make a full circle is $2\pi$ radians since the full circumference of a circle is $2\pi r$. Hence,
$2\pi^c =360^\circ$
Radians are sometimes indicated with a small $^c$ just like degrees are indicated with a $^\circ$. However, when working with radians, we often leave this off and the reader is expected to recognise which angle measurement is in use. We can now solve trigonometric equations giving answers in radians. See Example 1.
Arc Length
When the angle of a sector is given as $\theta$ radians, we can define the arc length to be $r\theta$, where $r$ is the radius. This follows from the definition above. Note that the full circle makes an angle of $2\pi $ radians and we have the part of the circumference that subtends from an angle of $\theta$. It follows that the proportion of the circle we have is $\frac{\theta}{2\pi}$. Multiplying this fraction by the full circumference will hence give us the length of the arc: $\frac{\theta}{2\pi}\times 2\pi r=r\theta$. See Examples 2 and 3.
Arc length = $r\theta$
The arc seen here is known as a minor arc. Note that we refer to arcs that are longer than half a circumference as major arcs.
Area of a Sector
When the angle at the centre of a circle is given as $\theta$ radians, we can define the area of a sector to be $\frac{1}{2}r^2\theta$, where $r$ is the radius. This also follows from the definition of radians above. Note that the full circle makes an angle of $2\pi $ radians and we have the part of the circumference that subtends from an angle of $\theta$. It follows that the proportion of the circle we have is $\frac{\theta}{2\pi}$. Multiplying this fraction by the full area of the circle will hence give us the area of the sector: $\frac{\theta}{2\pi}\times \pi r^2=\frac{1}{2}r^2\theta$. See Examples 2 and 3.
Area of Sector = $\frac{1}{2}r^2\theta$
We can use the area of a sector to find the area of a segment (video coming soon).
Examples using Radians
Example 1
Evaluate the following without a calculator:
- $\sin\left(\frac{\pi}{2}\right)$
- $\cos(\pi)$
- $\tan\left(\frac{\pi}{4}\right)$
- $\text{cosec}\left(\frac{3\pi}{4}\right)$
- $\text{sec}\left(\frac{5\pi}{4}\right)$
- $\text{cotan}\left((2n+1)\pi/2\right)$ where $n\in{\mathbb Z}$
See more on reciprocal trigonometric functions first.
Example 2
Find the arc length and area of this sector:
Example 3
The area of a sector with radius $4cm$ has an area of $10\pi$. Find the arc length.