Radians

This chapter introduces radians, an alternative to degrees for measuring angles. When you have completed it, you should
1. know how to convert from degrees to radians and vice versa
2. be able to use the formula rθ for the length of a circular arc, and 1/2r^2θ for the area of a circular sector
3. be able to solve trigonometric equations with roots expressed in radians.





Suppose that you were meeting angles for the first time, and that you were asked to suggest a unit for measuring them. It seems highly unlikely that you would suggest the degree, which was invented by the Babylonians in ancient times. The full circle, or the right angle, both seem more natural units.

However, the unit used in modern mathematics is the radian. This is particularly useful in differentiatig trigonometric functions, as you will see if you go on to deeper syllabus.

In a circle of radius 1 unit, radii joining the centre o to the ends of an arc of length 1 unit form an angle called 1 radian. The abbreviation for radian is rad.

You can see immediately from this definition that there are 2π radians in 360°. This leads to the following conversion rule for radians to degrees and vice versa.

              π rad = 180°


You could calculate that 1 radian is equal to 57.295 ...°, but no one uses this conversion. It is simplest to remember that π rad = 180°, and use this conversion between radians and degrees.

You can set your calculator to radian mode, and then work entirely in radians.
You might find on your calculator another unit for angle called the 'grad'; there are 100 grads to the right angle. Grads will not be used in this course.

Ex :
Convert 40° to radians, leaving your answer as a multiple of π.
       Since 40° is 2/9 of 180°, 40° = 2/9π rad.

It is worthwhile learning a few common conversions, so that you can think in both radians and degrees. For examples, you should know and recognise the following conversions :

180° = rad
90°   = 1/2π rad
45°   = 1/4π rad
30°   = 1/6π rad
60°   = 1/3π rad.




Formula :


           arc length    s = rθ
  

Area of circular sector     

 where r= radius ; θ = angle in rad


Note : no units are given in the formulae above. The units are appropriate units associated with the length; for instance, length in m and area in m^2.



 Example :

Find the perimeter and the area of the segment cut off by a chord PQ of length 8cm from a circle centre O and radius 6cm. Give you answers correct to 3 significant figures.

In problems of this type, it is helpful to start by thinking about the complete sector OPQ, rather than just the shaded segment.

The perimeter of the segment consists of two parts, the straight part of length 8cm, and the curved part; to calculate the length of the curved part you need to know the angle POQ.


Call this angle θ . As triangle POQ is isosceles, a perpendicular drawn from O to PQ bisects both PQ and angle POQ.


    sin 1/2θ = 4/6 = 0.6666 ... , so 1/2θ = 0.7297 ... and θ = 1.459 ...


make sure your calculator is in radian mode.

Then the perimeter d cm is given by d = 8 + 6θ = 16.756 ... ; the perimeter is 16.8cm, correct to 3 sf.

To find the area of the segment, you need to find the area of the sector OPQ, and then subtract the area of the triangle OPQ. Using the formula 1/2 bc sin A for the area of triangle, the area of the triangle POQ is given by 1/2 r^2 sin θ. Thus the area in cm^2 of the shaded region is 

The area is 8.38cm^2 , correct to 3 sf.



Solving trigonometric equations using radians


Ex : 
Solve the equation cos θ = - 0.7, giving all the roots in the interval 0 ≤ θ ≤ 2π, correct to 2 decimal places.


Step 1 : press your calculator  cos^ -1 0.7, ignore the sign .
You will come out with the answer θ = 0.795 rad .

Step 2 : draw this out.


 where a = all ; s = sin ; t = tan ; c = cos.

You have to remember this , this is the way i memorized it  ASTC = All School Teachers Crazy




Step 3 :
Since cos θ = - 0.7, so it is in second and third quadrants, S and T( because of the negative sign)
 
π - 0.795 = 2.35
π + 0.795 = 3.94

 The roots of the equation cos θ = - 0.7 are 2.35 and 3.94.

Note : You must beware of the interval given, if it asks you to give the interval  0 ≤ θ ≤ π, you cannot include 3.94 in your answer , because it is greater than π. So your answer should be 2.35 only.
 You can also solve the equation is degree, then convert the roots to radian for your final answer. But it is more convenient if you convert it to radian at first. 


More exercises will be uploaded soon .

Posted in: arc length,area of sector,circular arc,degree,radian,trigonometric equation