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# Six Trigonometric Functions and Radians

## Degrees versus radians and calculator modes.

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Practice Six Trigonometric Functions and Radians
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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 sinπ6\begin{align*}\sin \frac{\pi}{6}\end{align*}. You are about to start converting the measurements in radians into degrees when you wonder if it might be possible to just take the values of the functions directly.

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 sinπ6\begin{align*}\sin \frac{\pi}{6}\end{align*} directly.

### 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 sinπ6\begin{align*}\sin \frac{\pi}{6}\end{align*}, that is still sin30\begin{align*}\sin 30^\circ\end{align*} and the answer is still 12\begin{align*}\frac{1}{2}\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 12\begin{align*}\frac{1}{2}\end{align*}. In fact, as you may have suspected, the calculator is interpreting this as 30 radians. In this case, changing the mode to degrees and recalculating will give the expected result.

Scientific calculators will usually have a 3-letter display that shows either DEG or RAD to tell you which mode the calculator is in.

#### Example A

Find tan3π4\begin{align*}\tan \frac{3\pi}{4}\end{align*}.

Solution: If needed, convert 3π4\begin{align*}\frac{3\pi}{4}\end{align*} to degrees. Doing this, we find that it is 135\begin{align*}135^\circ\end{align*}. So, this is tan135\begin{align*}\tan 135^\circ\end{align*}, which is -1.

#### Example B

Find the value of cos11π6\begin{align*}\cos \frac{11\pi}{6}\end{align*}.

Solution: If needed, convert 11π6\begin{align*}\frac{11\pi}{6}\end{align*} to degrees. Doing this, we find that it is 330\begin{align*}330^\circ\end{align*}. So, this is cos330\begin{align*}\cos 330^\circ\end{align*}, which is 32\begin{align*}\frac{\sqrt{3}}{2}\end{align*}.

#### Example C

Convert 1 radian to degree measure.

Solution: Many students get so used to using π\begin{align*}\pi\end{align*} in radian measure that they incorrectly think that 1 radian means 1π\begin{align*}1\pi\end{align*} radians. While it is more convenient and common to express radian measure in terms of π\begin{align*}\pi\end{align*}, don’t lose sight of the fact that π\begin{align*}\pi\end{align*} radians is a number. It specifies an angle created by a rotation of approximately 3.14 radius lengths. So 1 radian is a rotation created by an arc that is only a single radius in length.

So 1 radian would be 180π\begin{align*}\frac{180}{\pi}\end{align*} degrees. Using any scientific or graphing calculator will give a reasonable approximation for this degree measure, approximately 57.3\begin{align*}57.3^\circ\end{align*}.

### Guided Practice

1. Using a calculator, find the approximate degree measure (to the nearest tenth) of the angle expressed in radians:

6π7\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:

20π11\begin{align*}\frac{20\pi}{11}\end{align*}

3. Gina wanted to calculate the sin210\begin{align*}\sin 210^\circ\end{align*} and got the following answer on her calculator:

Fortunately, Kylie saw her answer and told her that it was obviously incorrect.

2. Explain what she did wrong.

Solutions:

1. 154.3\begin{align*}154.3^\circ\end{align*}

2. 327.3\begin{align*}327.3^\circ\end{align*}

3. The correct answer is 12\begin{align*}-\frac{1}{2}\end{align*}. Her calculator was is the wrong mode and she calculated the sine of 210 radians.

### Concept Problem Solution

As you have learned in this Concept, the sinπ6\begin{align*}\sin \frac{\pi}{6}\end{align*} is the same as sin30\begin{align*}\sin 30^\circ\end{align*}, which equals 12\begin{align*}\frac{1}{2}\end{align*}. You could find this either by converting π6\begin{align*}\frac{\pi}{6}\end{align*} to degrees, or by using your calculator with angles entered in radians.

### Explore More

Using a calculator, find the approximate degree measure (to the nearest tenth) of the angle expressed in radians.

1. 4π7\begin{align*}\frac{4\pi}{7}\end{align*}
2. 5π6\begin{align*}\frac{5\pi}{6}\end{align*}
3. 8π11\begin{align*}\frac{8\pi}{11}\end{align*}
4. 5π3\begin{align*}\frac{5\pi}{3}\end{align*}
5. 8π3\begin{align*}\frac{8\pi}{3}\end{align*}
6. 7π4\begin{align*}\frac{7\pi}{4}\end{align*}
7. 12π5\begin{align*}\frac{12\pi}{5}\end{align*}

Find the value of each using your calculator.

1. sin3π2\begin{align*}\sin \frac{3\pi}{2}\end{align*}
2. cosπ2\begin{align*}\cos \frac{\pi}{2}\end{align*}
3. tanπ6\begin{align*}\tan \frac{\pi}{6}\end{align*}
4. sin5π6\begin{align*}\sin \frac{5\pi}{6}\end{align*}
5. tan4π3\begin{align*}\tan \frac{4\pi}{3}\end{align*}
6. cot7π3\begin{align*}\cot \frac{7\pi}{3}\end{align*}
7. sec11π6\begin{align*}\sec \frac{11\pi}{6}\end{align*}
8. Do you think radians will always be written in terms of π\begin{align*}\pi\end{align*}? Is it possible to have, for example, exactly 2 radians?