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.
Always check to see that your calculator is in the right mode, either degree or radians, when doing trigonometric calculations.
On the top of your calculator screen, it may say "D" for degree mode or "R" for radian mode.
Use the following applet to explore the values of sine, cosine and tangent function when the units are in radians or in degrees.
|
|
Practice questions
Question 1
Evaluate $\sin\frac{35\pi}{16}$sin35π16 correct to two decimal places.
Question 2
Evaluate $\cos6.87$cos6.87, where the angle given is measured in radians. Give your answer correct to two decimal places.
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
Find the exact value of $\sin\frac{\pi}{3}$sinπ3.
Question 4
Evaluate $\sin\frac{\pi}{6}\cos\frac{\pi}{4}$sinπ6cosπ4, leaving your answer in exact form with a rational denominator.
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
Find the reference angle for $\frac{2\pi}{3}$2π3.
Question 6
Find the exact value of $\cos\left(-\frac{5\pi}{3}\right)$cos(−5π3)
Question 7
By rewriting each ratio in terms of the related acute angle, evaluate the expression:
$\frac{\left(\sin\frac{2\pi}{3}\right)\left(\cos\frac{2\pi}{3}\right)\left(\tan\frac{3\pi}{4}\right)}{\tan\left(-\frac{\pi}{4}\right)}$(sin2π3)(cos2π3)(tan3π4)tan(−π4)
Give your answer in rationalised form.