2.3: Six Trigonometric Functions and Radians
While working in your math class one day, you are given a sheet of values in radians and asked to find the various trigonometric functions of them, such as sine, cosine, and tangent. The first question asks you to find the \begin{align*}\sin \frac{\pi}{6}\end{align*}
Do you think this is possible? As it turns out, it is indeed possible to apply trig functions to measurements in radians. Here you'll learn to do just that.
At the end of this Concept, you'll be able to compute \begin{align*}\sin \frac{\pi}{6}\end{align*}
Watch This
James Sousa: Determine Exact Trig Function Values With the Angle in Radians Using the Unit Circle
Guidance
Even though you are used to performing the trig functions on degrees, they still will work on radians. The only difference is the way the problem looks. If you see \begin{align*}\sin \frac{\pi}{6}\end{align*}
Most scientific and graphing calculators have a MODE setting that will allow you to either convert between the two, or to find approximations for trig functions using either measure. It is important that if you are using your calculator to estimate a trig function that you know which mode you are using. Look at the following screen:
If you entered this expecting to find the sine of 30 degrees you would realize that something is wrong because the answer should be \begin{align*}\frac{1}{2}\end{align*}
Scientific calculators will usually have a 3letter display that shows either DEG or RAD to tell you which mode the calculator is in.
Example A
Find \begin{align*}\tan \frac{3\pi}{4}\end{align*}
Solution: If needed, convert \begin{align*}\frac{3\pi}{4}\end{align*}
Example B
Find the value of \begin{align*}\cos \frac{11\pi}{6}\end{align*}
Solution: If needed, convert \begin{align*}\frac{11\pi}{6}\end{align*}
Example C
Convert 1 radian to degree measure.
Solution: Many students get so used to using \begin{align*}\pi\end{align*}
\begin{align*}\text{radians} \times \frac{180}{\pi}=\text{degrees}\end{align*}
So 1 radian would be \begin{align*}\frac{180}{\pi}\end{align*}
Vocabulary
Radian: A radian (abbreviated rad) is the angle created by bending the radius length around the arc of a circle.
Guided Practice
1. Using a calculator, find the approximate degree measure (to the nearest tenth) of the angle expressed in radians:
\begin{align*}\frac{6\pi}{7}\end{align*}
2. Using a calculator, find the approximate degree measure (to the nearest tenth) of the angle expressed in radians:
\begin{align*}\frac{20\pi}{11}\end{align*}
3. Gina wanted to calculate the \begin{align*}\sin 210^\circ\end{align*}
Fortunately, Kylie saw her answer and told her that it was obviously incorrect.
 Write the correct answer, in simplest radical form.
 Explain what she did wrong.
Solutions:
1. \begin{align*}154.3^\circ\end{align*}
2. \begin{align*}327.3^\circ\end{align*}
3. The correct answer is \begin{align*}\frac{1}{2}\end{align*}
Concept Problem Solution
As you have learned in this Concept, the \begin{align*}\sin \frac{\pi}{6}\end{align*}
Practice
Using a calculator, find the approximate degree measure (to the nearest tenth) of the angle expressed in radians.

\begin{align*}\frac{4\pi}{7}\end{align*}
4π7 
\begin{align*}\frac{5\pi}{6}\end{align*}
5π6  \begin{align*}\frac{8\pi}{11}\end{align*}
 \begin{align*}\frac{5\pi}{3}\end{align*}
 \begin{align*}\frac{8\pi}{3}\end{align*}
 \begin{align*}\frac{7\pi}{4}\end{align*}
 \begin{align*}\frac{12\pi}{5}\end{align*}
Find the value of each using your calculator.
 \begin{align*}\sin \frac{3\pi}{2}\end{align*}
 \begin{align*}\cos \frac{\pi}{2}\end{align*}
 \begin{align*}\tan \frac{\pi}{6}\end{align*}
 \begin{align*}\sin \frac{5\pi}{6}\end{align*}
 \begin{align*}\tan \frac{4\pi}{3}\end{align*}
 \begin{align*}\cot \frac{7\pi}{3}\end{align*}
 \begin{align*}\sec \frac{11\pi}{6}\end{align*}
 Do you think radians will always be written in terms of \begin{align*}\pi\end{align*}? Is it possible to have, for example, exactly 2 radians?
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