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?)

radians

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

radiansWhen 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.

Examples

Evaluate the following without a calculator:

  1. \sin\left(\frac{\pi}{2}\right)
  2. \cos(\pi)
  3. \tan\left(\frac{\pi}{4}\right)
  4. \text{cosec}\left(\frac{3\pi}{4}\right)
  5. \text{sec}\left(\frac{5\pi}{4}\right)
  6. \text{cotan}\left((2n+1)\pi/2\right) where n\in{\mathbb Z}

See more on reciprocal trigonometric functions first.

Solution:

  1. \sin\left(\frac{\pi}{2}\right)=\sin(90^\circ)=1
  2. \cos(\pi)=\cos(180^\circ)=-1
  3. \tan\left(\frac{\pi}{4}\right)=\tan(45^\circ)=1
  4. \text{cosec}\left(\frac{3\pi}{4}\right)=\frac{1}{\sin(135^\circ)}=\frac{1}{\frac{\sqrt{2}}{2}}=\sqrt{2}
  5. \text{sec}\left(\frac{5\pi}{4}\right)=\frac{1}{\cos(225^\circ)}=\frac{1}{-\frac{\sqrt{2}}{2}}=-\sqrt{2}
  6. \text{cotan}\left((2n+1)\pi/2\right)=0 for odd multiples of \pi/2 because this is where tan has asymptotes.

Find the arc length and area of this sector:radians

Solution:

Arc Length = r\theta =10\times \frac{\pi}{4}=\frac{5\pi}{2}

Area of Sector = \frac{1}{2}r^2\theta =\frac{1}{2}\times 100\times \frac{\pi}{4}=\frac{25\pi}{2}

The area of a sector with radius 4cm has an area of 10\pi. Find the arc length.

Solution:

We must find the angle of the sector first. Solving \frac{1}{2}r^2\theta=10\pi with r=4 gives \theta=\frac{5\pi}{4}. It follows that the arc length is r\theta =4\times \frac{5\pi}{4}=5\pi. Note that this is a major arc (more than half a circumference) as the angle is greater than \pi.

AS Maths Trigonometry

A2 Maths Trigonometry