06Aradians

toc = Radians & the Unit Circle =

The size of an angle can be measured in: or
 * ** degrees ** (degrees, minutes, seconds)
 * ** radians **.

Definition of a Radian

 * 1) We draw a circle with a radius of 1 unit (the Unit Circle).
 * 2) We make an arc along the circumference with an arc-length of 1 unit (the same as the radius).
 * 3) the angle formed at the centre of the circle by that arc is defined as 1 radian.
 * Notice that this definition works regardless of the size of the “unit” being used.


 * From this, we can see that for a unit circle, the size of any angle (in radians) is equal to the length (in units) of the arc formed by that angle.

Notation

 * We indicate the angle is measured in __ degrees __ with a small circle (eg 20º).
 * We indicate the angle is measured in __ radians __ with a small “c” (eg 2.3 c ).


 * It is very important to use the correct notation. There is an enormous difference between 1º and 1 c, so failure to indicate which measurement system is being used can result in big errors (as well as losing marks).

Radians v. Degrees

 * We know that for any circle, the circumference is C = 2 p r.
 * Hence, for the unit circle (radius = 1) the circumference is C = 2 p.


 * With a unit circle, the size of the angle in radians is equal to the size of the arc length formed by that angle.
 * So for a full revolution (ie 360º), the angle in radians must be the same as the circumference of the circle (ie 2 p )


 * Hence
 * 2 p c = 360º
 * or
 * p c = 180º

Common Exact Angles

 * These are common angles expressed as a fraction of p.


 * Fractions of p are exact values which are preferred to approximate (decimal) answers when possible.


 * Note that when writing an angle as a fraction of p it is assumed to be in radians so we can usually leave off the (c) notation.



Converting Radians to Degrees

 * Given that p c = 180º, we can divide both sides by p to get:

math . \qquad 1^c=\dfrac{180^\circ}{\pi} math


 * So to convert a given number of radians into degrees,

math . \qquad \text{MULTIPLY BY } \dfrac{180^\circ}{\pi} math


 * ** If the angle is expressed as a fraction of p, this is equivalent to replacing p with 180º in the fraction. **

** Example 1 **

 * Convert into degrees

math \\ . \qquad \textbf{ (a) } \quad \dfrac{2\pi}{5} = \dfrac{2 \times 180^\circ}{5} = 72^\circ \\. \\ . \qquad \textbf{ (b) } \quad 2.3^c = 2.3 \times \dfrac{180^\circ}{\pi} = 131.78^\circ math

Converting Degrees to Radians

 * Given that p c = 180º, we can divide both sides by 180 to get:

math . \qquad 1^\circ = \dfrac{\pi}{180} math


 * So to convert a given number of degrees into radians,

math . \qquad \text{MULTIPLY BY } \dfrac{\pi}{180} math

** Example 2 **

 * Convert into radians (as a fraction of p )

math . \qquad \textbf{ (a)} \quad 80^\circ = 80 \times \dfrac{\pi}{180} = \dfrac{4\pi}{9} math

math . \qquad \textbf{ (b) } \quad 212.38^\circ = 212.38 \times \dfrac{\pi}{180} = 3.71^c math
 * Convert into radians (to 2 decimal places)

Radians on the CAS calculator

 * On the Classpad, you can select between Degrees and Radians mode in the bottom row of the screen.


 * A value entered will be assumed to be in the same mode as the calculator setting so it is not necessary to add the appropriate symbol.
 * in Degrees mode, entering sin(30) will be assumed to mean sin(30º)
 * in Radians mode, entering cos(3.142) will be assumed to mean cos(3.142 c )


 * You can override the setting and enter an angle in the other form by including the symbol
 * Classpad uses a little “r” symbol for radians


 * The ** º ** and ** r ** symbols are in the virtual keyboard, ** mth ** tab, ** TRIG ** page

media type="custom" key="8127530"

Converting b/w Radians and Degrees on the Classpad
media type="custom" key="8127544" align="right"
 * We can change degrees to radians by
 * changing to __ **radian** __ __ **mode** __ and entering the angle followed by the degrees sign(º)
 * {choose between media type="custom" key="8127572" or media type="custom" key="8127550" (fraction of **p** )}


 * We can change radians to degrees by
 * changing to __ **degrees mode** __ and entering the angle followed by the radians sign(r)
 * {enter either a media type="custom" key="8127554" or a media type="custom" key="8127558" of **p** }

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