Introduction

We've seen how to apply trigonometric ratios to angles given in degrees, but we can also apply them to angles given in radians.

For example, let's consider the expression \cos({\color{red}1}). Note that \color{red}1 is written without any special symbols, like {}^\circ (degrees). This indicates that the angle \color{red}1 is given in radians. Here, \color{red}1 reads as "one radian."

If we were given \cos({\color{blue}1^\circ}) instead, with the degree sign {}^\circ , that would mean we want the cosine of "one degree."

The simplest way to find the actual value of \cos(1) would be using a calculator. This gives \cos(1) = 0.540\,302\ldots \approx 0.540 rounded to 3 decimal places.

Watch out! Don't forget to set your calculator into the "radians" mode before computing the trigonometric function above.

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Example: Finding a Trigonometric Ratio

Given that x=0.3 radians, what is \sin{x} rounded to 3 decimal places?

EXPLANATION

Note the argument of the sine is given in radians. Using a calculator in "radians" mode, we obtain \sin(0.3) = 0.295\,520\ldots \approx 0.296 rounded to 3 decimal places.

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Practice: Finding a Trigonometric Ratio

Given that $x=0.7$ radians, what is $\cos{x}$ rounded to $3$ decimal places?

a
 $0.765$
b
 $0.999$
c
 $0.644$
d
 $0.342$
e
 $0.012$
Practice: Finding a Trigonometric Ratio

Given that $x=1.1$ radians, what is $\tan{x}$ rounded to $3$ decimal places?

a
 $0.891$
b
 $1.965$
c
 $0.019$
d
 $0.044$
e
 $0.454$
Example: Finding an Inverse Trigonometric Ratio

Rounded to three decimal places, what is \arcsin(0.75) in radians?

EXPLANATION

Note that we need the answer in radians. Using a calculator in "radians" mode, we obtain \arcsin(0.75) = 0.848\,062\ldots \approx 0.848 \, \textrm{radians} rounded to 3 decimal places.

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Practice: Finding an Inverse Trigonometric Ratio

Rounded to three decimal places, what is $\arctan(0.8)$ in radians?

a
 $0.540$
b
 $1.446$
c
 $0.675$
d
 $1.558$
e
 $0.381$
Practice: Finding an Inverse Trigonometric Ratio

Rounded to three decimal places, what is $\arccos(0.4)$ in radians?

a
 $0.927$
b
 $1.369$
c
 $0.644$
d
 $1.159$
e
 $1.047$
Trigonometric Ratios With Angles in Radians Written As Multiples of Pi

Another way to compute a trigonometric ratio of an angle, say \sin\theta where \theta is measured in radians, is to convert the angle into degrees first and then compute the trigonometric ratio of the equivalent angle in degrees.

Let's consider the value of \sin \left(\dfrac{\pi}{10}\right). Since the angle is a multiple of \pi, we can easily convert the measure in radians to an equivalent measure in degrees. This gives

\left(\dfrac{\pi}{10}\right) \left(\dfrac {180^\circ} {\pi}\right) = \dfrac {180^\circ} {10} = 18^\circ.

Now, using a calculator in "degrees" mode, we obtain \begin{align*} \sin \left(\dfrac{\pi}{10}\right) &= \sin (18^\circ) \\[2pt] &= 0.309\,016\ldots \\[2pt] &\approx 0.309 \end{align*} rounded to 3 decimal places.

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Example: Finding a Trigonometric Ratio of an Acute Special Angle

Evaluate \sin \left(\dfrac{\pi}{6}\right) by converting the angle to degrees.

EXPLANATION

First, we convert \dfrac{\pi}{6} radians to an equivalent measure in degrees. This gives

\left(\dfrac{\pi} 6\right) \left(\dfrac {180^\circ} {\pi}\right) = \dfrac {180^\circ} {6} = 30^\circ.

Therefore, \sin \left(\dfrac{\pi}{6}\right) = \sin 30^\circ.

Finally, since we know that \sin 30^\circ = \dfrac{1}{2}, we obtain \sin\left( \dfrac{\pi}{6}\right) = \dfrac{1}{2}.

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Practice: Finding a Trigonometric Ratio of an Acute Special Angle

$\tan \left(\dfrac{\pi}{4}\right) = $

a
 $\dfrac{\sqrt{3}}{2}$
b
 $\dfrac{3}{2}$
c
 $\dfrac{\sqrt{2}}{2}$
d
 $1$
e
 $\dfrac{1}{2}$
Practice: Finding a Trigonometric Ratio of an Acute Special Angle

$\cos \left(\dfrac{\pi}{3}\right) = $

a
 $1$
b
 $\sqrt{3}$
c
 $\dfrac{\sqrt{2}}{2}$
d
 $\dfrac{1}{2}$
e
 $\dfrac{\sqrt{3}}{2}$
Example: Identifying True Statements

Is the following statement true or false?

\sin30^\circ \:\overset{\large\color{red}?}{=}\: \sin \left( \dfrac16 \right).

EXPLANATION

The statement is false. Converting 30^\circ from degrees to radians, we get

30^\circ= 30\cdot \dfrac{\pi}{180} = \dfrac{\pi}{6} \, \textrm{radians}.

The statement is false because \dfrac{\pi}{6} \neq \dfrac{1}6.

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Practice: Identifying True Statements

Which of the following statements are true?

  1. $\tan 5^\circ = \tan \left(\dfrac{\pi}{36}\right)$
  2. $\tan 10^\circ = \tan \left(\dfrac{\pi}{18}\right)$
  3. $\tan 15^\circ = \tan \left(\dfrac{\pi}{9}\right)$
a
 I, II and III
b
 II and III only
c
 I and II only
d
 I only
e
 I and III only
Practice: Identifying True Statements

Which of the following statements are true?

  1. $\sin 20^\circ = \sin \left(\dfrac{\pi}{9}\right)$
  2. $\sin 50^\circ = \sin \left(\dfrac{2\pi}{9}\right)$
  3. $\sin 80^\circ = \sin \left(\dfrac{4\pi}{9}\right)$
a
 II only
b
 I and II only
c
 I and III only
d
 II and III only
e
 I, II and III
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