To find the values of these trigonometric functions, we need to carefully analyze each angle and its corresponding function. Let's proceed step-by-step.
### Calculating [tex]\(\tan \left(-\frac{7 \pi}{6}\right)\)[/tex]
1. Identify the Reference Angle:
- The angle [tex]\(-\frac{7\pi}{6}\)[/tex] is in the third quadrant because the angle is more than [tex]\(-\pi\)[/tex] and less than [tex]\(-3\pi/2\)[/tex].
2. Simplify the Angle:
- We can add [tex]\(2\pi\)[/tex] to normalize the angle within a standard cycle:
[tex]\[
-\frac{7\pi}{6} + 2\pi = -\frac{7\pi}{6} + \frac{12\pi}{6} = \frac{5\pi}{6}
\][/tex]
3. Evaluate Tangent in the Third Quadrant:
- The reference angle is [tex]\(\frac{\pi}{6}\)[/tex] and the tangent of [tex]\(\pi/6\)[/tex] is [tex]\(1/\sqrt{3}\)[/tex], but we need to consider the signs.
- In the third quadrant, tangent is positive:
[tex]\[
\tan\left(-\frac{7\pi}{6}\right) = \tan\left(\frac{5\pi}{6}\right) = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}
\][/tex]
So,
[tex]\[
\tan \left(-\frac{7 \pi}{6}\right) = -\frac{\sqrt{3}}{3}
\][/tex]
### Calculating [tex]\(\csc \left(\frac{\pi}{4}\right)\)[/tex]
1. Definition of Cosecant:
- [tex]\(\csc(x) = \frac{1}{\sin(x)}\)[/tex]
2. Sine of [tex]\(\frac{\pi}{4}\)[/tex]:
[tex]\[
\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}
\][/tex]
3. Calculate Cosecant:
[tex]\[
\csc\left(\frac{\pi}{4}\right) = \frac{1}{\sin\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}
\][/tex]
So,
[tex]\[
\csc\left(\frac{\pi}{4}\right) = \sqrt{2}
\][/tex]
### Calculating [tex]\(\cos \left(\frac{17 \pi}{3}\right)\)[/tex]
1. Simplify the Angle to a Standard Range:
- Divide [tex]\(\frac{17\pi}{3}\)[/tex] by [tex]\(2\pi\)[/tex]:
[tex]\[
\frac{17\pi}{3} \text{ mod } 2\pi = \frac{17\pi}{3} - 2\pi \left\lfloor \frac{17\pi}{3} / 2\pi \right\rfloor
\][/tex]
- The integer division of [tex]\(\frac{17\pi}{3}\)[/tex] by [tex]\(2\pi\)[/tex] is:
[tex]\[
\frac{17}{3 \cdot 2} = \frac{17}{6}
\][/tex]
- [tex]\(\left\lfloor \frac{17}{6} \right\rfloor = 2\)[/tex]
- Therefore, the angle is:
[tex]\[
\frac{17\pi}{3} - 2\pi \cdot 2 = \frac{17\pi}{3} - \frac{12\pi}{3} = \frac{5\pi}{3}
\][/tex]
2. Evaluate Cosine in the First Quadrant:
- The reference angle for [tex]\(\frac{5\pi}{3}\)[/tex] is:
[tex]\[
2\pi - \frac{5\pi}{3} = \frac{\pi}{3}
\][/tex]
- The cosine of [tex]\(\frac{\pi}{3}\)[/tex] is:
[tex]\[
\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}
\][/tex]
So,
[tex]\[
\cos\left(\frac{17\pi}{3}\right) = \frac{1}{2}
\][/tex]
### Final Answer
[tex]\[
\begin{array}{l}
\tan \left(-\frac{7 \pi}{6}\right)=-\frac{\sqrt{3}}{3} \\
\csc \left(\frac{\pi}{4}\right)=\sqrt{2} \\
\cos \left(\frac{17 \pi}{3}\right)=\frac{1}{2}
\end{array}
\][/tex]
