So far we have expressed angles in degrees, but in mathematics angles are often expressed in radians. To introduce radians we will use a circle with radius #1#. We call this the unit circle.
The unit circle is a circle with origin center #\rv{0,0}# and radius #1#.
The point #P=\rv{\blue{x_P}, \purple{y_P}}# starts at #\rv{1,0}# and moves counterclockwise across the unit circle. The angle of rotation is called #\green{\alpha}#.
Therefore #\sin(\green{\alpha})=\purple{y_P}# and #\cos(\green{\alpha})=\blue{x_P}#.
In this way we can also define angles greater than #90^\circ# degrees with the sine and the cosine.
A very useful rule for every angle #\green{\alpha}# in degrees, is the following:
\[\sin(\green{\alpha})^2+\cos(\green{\alpha})^2=1\]
By using the unit circle and the Pythagorean theorem we can deduce this rule. In the unit circle we have the right-angled triangle with legs #\purple{y_p}=\sin(\green{\alpha})# and #\blue{x_p}=\cos(\green{\alpha})#, and the hypotenuse with length #1#. By the Pythagorean theorem we now have
\[\purple{y_p}^2+\blue{x_p}^2=1\]
and thus
\[\sin(\green{\alpha})^2+\cos(\green{\alpha})^2=1\]
Examples
\[\begin{array}{rcl}\sin(\green{60^\circ})^2+\cos(\green{60^\circ})^2&=& \\ \left(\purple{\frac{\sqrt{3}}{2}}\right)^2 + \left(\blue{\frac{1}{2}}\right)^2 &=&\\ \frac{3}{4}+\frac{1}{4}&=&\\1 \\ \\ \sin(\green{225^\circ})^2+\cos(\green{225^\circ})^2&=&\\ \left(\purple{-\frac{\sqrt{2}}{2}}\right)^2 + \left(\blue{-\frac{\sqrt{2}}{2}}\right)^2&=&\\\frac{1}{2}+\frac{1}{2}&=&\\1 \end{array}\]
With this unit circle we can now express the size of the angle in radians.
The size of angle #\green{\alpha}# in the unit circle in radians is the length of the arc on the unit circle.
The length of the entire arc is #2 \pi#. Therefore, an angle of #360^\circ# is equal to #2 \pi# radians.
An angle #\green{\alpha}# in degrees measures #\tfrac{\green{\alpha}}{180}\cdot\pi# radians.
The angle of #30# degrees measures #\tfrac{\green{30}}{180}\cdot\pi = \tfrac 16 \pi#.
An angle #\green{\alpha}# in radians measures #180\cdot \tfrac{\green{\alpha}}{\pi}# degrees.
The angle #\pi# radians measures #180 \cdot \tfrac{\green{\pi}}{\pi} = 180# degrees.
There are also angles greater than #360^\circ# or #2 \pi#. These angles are more than a full rotation on the unit circle. The length of the arc is then \[\text{length of arc}=2 \pi \cdot \text{number of full rotations}+\text{angle on unit circle}\]
The cosine and sine of angles of which we add a multiple of #2 \pi# are therefore equal. Therefore:
\[\begin{array}{c}\sin(\alpha+2 \pi)=\sin(\alpha)\\ \\ \cos(\alpha+2 \pi)=\cos(\alpha) \end{array}\]
The same applies to negative angles, only then we do full rotations backwards on the unit circle.
Example
\[\begin{array}{rcl}\cos(\tfrac{5}{2}\pi)&=&\cos(\tfrac{1}{2}\pi+2 \pi)\\ &=&\cos(\tfrac{1}{2}\pi)\\ &=&0 \\ \\ \sin(\tfrac{7}{3}\pi)&=&\sin(\tfrac{1}{3}\pi+2 \pi) \\&=& \sin(\tfrac{1}{3}\pi)\\ &=&\tfrac{1}{2}\sqrt{3}\end{array}\]
When we work with angles in radians using our calculator, we have to set it to radians.
Note that when we do so, your calculator will often give answers as decimal numbers, while we usually work with exact values. In the section Special values of trigonometric functions, we'll have a look at the most important values.
When calculating with radian we have, for every angle #\green{\alpha}#, the same rule \[\sin(\green{\alpha})^2+\cos(\green{\alpha})^2=1\]
We have the right-angled triangle with legs #\purple{y_p}=\sin(\green{\alpha})# and #\blue{x_p}=\cos(\green{\alpha})#, and the hypotenuse with length #1#. Again the rule follows by the Pythagorean theorem.
Therefore, it does not matter if we are calculating in degrees or in radians in order for the rule to hold.
Examples
\[\begin{array}{rcl}\sin(\green{\frac{1}{3}\pi})^2+\cos(\green{\frac{1}{3}\pi})^2&=& \\ \left(\purple{\frac{\sqrt{3}}{2}}\right)^2 + \left(\blue{\frac{1}{2}}\right)^2 &=&\\ \frac{3}{4}+\frac{1}{4}&=&\\1 \\ \\ \sin(\green{\frac{5}{4}\pi})^2+\cos(\green{\frac{5}{4}\pi})^2&=&\\ \left(\purple{-\frac{\sqrt{2}}{2}}\right)^2 + \left(\blue{-\frac{\sqrt{2}}{2}}\right)^2&=&\\\frac{1}{2}+\frac{1}{2}&=&\\1 \end{array}\]
The sine of an angle #\alpha#, in radians or in degrees, gives the #y#-coordinate of the point on the unit circle which has angle #\alpha# relative to the #x#-axis.
The cosine of an angle #\alpha#, in radians or in degrees, gives the #x#-coordinate of the point on the unit circle which has angle #\alpha# relative to the #x#-axis.
How many radians does an angle of #67# degrees measure?
Give your answer in the form of a decimal number with two decimal digits.
#1.17# radians
According to the theory, an angle of #\alpha# degrees measures exactly #\dfrac{\alpha\cdot\pi}{180}# radians.
In order to find the answer to the question, we enter #\alpha=67# into this expression:
\[
\dfrac{67\cdot \pi}{180}\approx 1.17\tiny.\]
Unit circle and angles in radians
