Sure, let's work through each step to determine the values for the six trigonometric functions for the point [tex]\((-2, -10)\)[/tex] on the terminal ray of angle [tex]\(\theta\)[/tex]. Here is a detailed step-by-step solution.
### Step 1: Calculate the Distance [tex]\( r \)[/tex]
First, we need to find the distance [tex]\( r \)[/tex] from the origin to the point [tex]\((-2, -10)\)[/tex]. This distance is the hypotenuse of the right triangle formed by the point and the origin. We use the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Substituting [tex]\( x = -2 \)[/tex] and [tex]\( y = -10 \)[/tex]:
[tex]\[
r = \sqrt{(-2)^2 + (-10)^2} = \sqrt{4 + 100} = \sqrt{104} \approx 10.198
\][/tex]
So, [tex]\( r \approx 10.198 \)[/tex].
### Step 2: Calculate [tex]\(\sin \theta\)[/tex]
[tex]\(\sin \theta\)[/tex] is given by:
[tex]\[
\sin \theta = \frac{y}{r} = \frac{-10}{10.198} \approx -0.981
\][/tex]
### Step 3: Calculate [tex]\(\cos \theta\)[/tex]
[tex]\(\cos \theta\)[/tex] is given by:
[tex]\[
\cos \theta = \frac{x}{r} = \frac{-2}{10.198} \approx -0.196
\][/tex]
### Step 4: Calculate [tex]\(\tan \theta\)[/tex]
[tex]\(\tan \theta\)[/tex] is given by:
[tex]\[
\tan \theta = \frac{y}{x} = \frac{-10}{-2} = 5
\][/tex]
### Step 5: Calculate [tex]\(\csc \theta\)[/tex]
[tex]\(\csc \theta\)[/tex] (the reciprocal of [tex]\(\sin \theta\)[/tex]) is given by:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-0.981} \approx -1.020
\][/tex]
### Step 6: Calculate [tex]\(\sec \theta\)[/tex]
[tex]\(\sec \theta\)[/tex] (the reciprocal of [tex]\(\cos \theta\)[/tex]) is given by:
[tex]\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{-0.196} \approx -5.099
\][/tex]
### Step 7: Calculate [tex]\(\cot \theta\)[/tex]
[tex]\(\cot \theta\)[/tex] (the reciprocal of [tex]\(\tan \theta\)[/tex]) is given by:
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{5} = 0.2
\][/tex]
### Summary of Results
So, summarizing all the calculated values, we have:
[tex]\[
\begin{aligned}
r &\approx 10.198 \\
\sin \theta &\approx -0.981 \\
\cos \theta &\approx -0.196 \\
\tan \theta &= 5 \\
\csc \theta &\approx -1.020 \\
\sec \theta &\approx -5.099 \\
\cot \theta &= 0.2 \\
\end{aligned}
\][/tex]
