Revision:OCR Core 2 - Radians

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Revision:OCR Core 2 - Radians

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TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > OCR Core 2 - Radians

1 9. Radians
2 Also See

9. Radians

Radians

Would one naturally choose the unit of the "degree" for the measurement of angle? It is rather unlikely, as having 360 of these "degrees" in a full circle seems somewhat arbitrary.

The radian is simply another unit of angle, which is particularly helpful in calculus involving trigonometric functions.

One must be aware of radians, and their implications.

The definition of a radius is simple. If one is to take a circle, and then take one radius, a radian is the angle formed between two radii who are separated (at the circumference) by the length of one radius.

As the circumference of a circle is $2 \pi r$ , there are $2 \pi$ radians in $360^{ \circ }$ .

Hence:

$180^{ \circ } = \pi \ rad$

Formulae in degrees and radians

One has seen the identities which regard the trigonometric functions. Identities which do not refer to an angle are unaffected by the unit of measure (as one might expect).

However, one must consider the identities such as:

$\sin{ \theta} \equiv \sin{ \left( 180^{ \circ } - \theta } \right)$

As this has an angle which is measured in degrees in it, the identity is affected, but only in that one must convert the angle:

$\sin{ \theta } \equiv \sin{ \pi - \theta }$

(Often there is no symbol for the radian, it is merely expected that if one encounters an angle which is not indicated to be measured in degrees, it is measured in radians).

Length of arc and area of sector

Radians have implications on the calculation of the length of an arc, and indeed the area of a sector.

One can calculate the length of a circular arc quite simply as a fraction of the circumference:

$Let \ \theta \ be \ the \ angle \ subtended \ by \ the \ arc \ at \ the \ centre \ of \ the \ circle$

$\frac{ \theta }{2 \pi } \times 2 \pi r = r \theta$

This simple formula can be used to calculate the length of any circular arc, where the angle is measured in radians.

The area of a sector is another question of a fraction of the area of the whole circle:

$\frac{ \theta }{2 \pi } \theta \pi r^{2} = \frac{1}{2}r^{2} \theta$ .

One may be asked to calculate the area within a specific part of the circle. This will involve the calculation of the area of a sector, and also the calculation of the area of a triangle (in general).

One should be careful to ensure that one has one's calculator in the "Radian" mode, however one can be sure that using the sine, cosine and other formulae that are non-unit-specific will produce correct answers (if used correctly).

Graphs of the trigonometric functions

One might be able to tell that there is no change in the shapes of the graphs of the trigonometric functions, however one must remember that there are radians on the x-axis, so, for instance, on the graph of the sine function, there is an x-axis intercept at $( \pi , \ 0 )$ .

It is not necessary to "re-learn" the properties of the trigonometric functions, one must simply realise that if one is aware of the common angles, and their radian equivalents, one can simply remember a single set of the properties, and convert the angles. There is no change to the trigonometry behind the properties, as radians are a unit. (Just remember to convert the angles).

Solving trigonometric equations using radians

There is no change in the method that one would use to solve the trigonometric equations were they in degrees. It is important that one remembers to convert the properties, and that it is not usually a good idea to convert from radians to degrees to solve the problem, and then back again. One should be able to use the radians rather like degrees.

Example

1. Solve the following (for $0 \le x \le 2 \pi$ ):

$\sin{2x - \pi} = \frac{1}{2}$ .

This is simply what one might do with any other trigonometric equation.

$y = 2x - \pi$

Calculate the bounds for "y":

$- \pi \le y \le 3 \pi$

Now:

$\sin^{-1}(y) = \frac{ \pi }{6}$

$\sin{ \theta } = \sin{ \pi - \theta}$

Hence:

$y = \frac{ \pi}{6}, \ \frac{5 \pi}{6}$

(Adding, or subtracting $2 \pi$ will make the answer above, or below the bounds).

Hence:

$x = \frac{7 \pi}{12}, \ \frac{11 \pi}{12}$

One should be vigilant of the more difficult questions, however, upon the subject of radians, one should simply be aware of the relatively small ramifications of their use.