5.08 Using radians

 

In this lesson, we will be taking the skills studied in chapter 6 and using radians as a measure of an angle instead of degrees. 

Radian mode on the calculator

To evaluate trigonometric functions that use radians on the calculator, we need to set the calculator to be in radian mode. This is very important for getting the correct answer when evaluating expressions such as $\sin3.2^c$sin3.2c where the angle is in radians.

Different scientific calculators will have their own instructions to do this. For the Casio fx-82AU Plus calculator, you can change the mode by pressing: shift - setup - 4.

                                                                            

Alternatively, we could convert the radian angle to degrees first to avoid having to change the calculator to radian mode. 

Practice questions

Question 1

Question 2

Exact trigonometric ratios in radians

In the last chapter we found the exact trigonometric ratios for angles $30^\circ$30°, $60^\circ$60° and $45^\circ.$45°.

We can now express these trigonometric ratios in radians rather than degrees:

$30^\circ=\frac{\pi}{6}$30°=π6​

$45^\circ=\frac{\pi}{4}$45°=π4​

$60^\circ=\frac{\pi}{3}$60°=π3​

The two triangles that give exact trigonometric ratios can be drawn using radians instead of degrees.

                                                                   

From the diagram we can read off the following function values:

 

$\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}$sinπ3​=√32​	$\sin\frac{\pi}{6}=\frac{1}{2}$sinπ6​=12​	$\sin\frac{\pi}{4}=\frac{1}{\sqrt{2}}$sinπ4​=1√2​
$\cos\frac{\pi}{3}=\frac{1}{2}$cosπ3​=12​	$\cos\frac{\pi}{6}=\frac{\sqrt{3}}{2}$cosπ6​=√32​	$\cos\frac{\pi}{4}=\frac{1}{\sqrt{2}}$cosπ4​=1√2​
$\tan\frac{\pi}{3}=\sqrt{3}$tanπ3​=√3	$\tan\frac{\pi}{6}=\frac{1}{\sqrt{3}}$tanπ6​=1√3​	$\tan\frac{\pi}{4}=1$tanπ4​=1

Practice questions

Question 3

Question 4

Angles of any magnitude using radians

The unit circle can be expressed in radians as shown in this diagram:

The steps in finding the value of a trigonometric function for any angle remain the same, i.e. identifying the quadrant, finding the related acute angle, and working out the trigonometric function for the related acute angle. It is just a matter of getting used to thinking in radians instead of degrees!

Practice questions

Question 5

Question 6

Question 7

Trigonometric graphs in radians

When sketching trigonometric functions using radians, we can simply change the values on the horizontal axis from degrees to radians. The domain also changes accordingly.

                               

 

Now that we are working in radians it is easier to graph other functions together with trigonometric functions in order to solve equations graphically. This is because we are no longer using degrees but radians on the $x$x-axis, meaning that other functions of $x$x are possible.

Worked example

Determine the number of solutions that satisfy the equation $2\sin x=x$2sinx=x.

Think: What does the graph of $y=2\sin x$y=2sinx and $y=x$y=x look like? The solutions to the equation will be the same as the points of intersection of the two graphs.

Do: Draw a number plane and mark a scale with $\frac{-\pi}{2}$−π2​, $-\pi$−π, $\frac{\pi}{2}$π2​, $\pi$πetc. and also approximate the positions of  $-1$−1,$-2$−2,$0$0,$1$1,$2$2, and $3$3 etc. (you can do this using your calculator: $pi\approx3.14$pi≈3.14).

Sketch the graph of $y=2\sin x$y=2sinx and $y=x$y=xand look for the number of times the graphs meet.

It is clear that the graphs meet at three points, therefore there are $3$3 solutions to the equation $2\sin x=x$2sinx=x.

Practice questions

Question 8

Question 9