To solve this problem, we need to find the exact values of [tex]\(\cos \theta\)[/tex], [tex]\(\sec \theta\)[/tex], and [tex]\(\cot \theta\)[/tex] for the point [tex]\((-3, 8)\)[/tex] which lies on the terminal side of the angle [tex]\(\theta\)[/tex].
1. Calculate the hypotenuse (radius) [tex]\(r\)[/tex]
The hypotenuse [tex]\(r\)[/tex] can be calculated using the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Here, [tex]\(x = -3\)[/tex] and [tex]\(y = 8\)[/tex]. Plugging in these values:
[tex]\[
r = \sqrt{(-3)^2 + 8^2} = \sqrt{9 + 64} = \sqrt{73}
\][/tex]
Therefore, [tex]\(r\)[/tex] is approximately 8.544.
2. Calculate [tex]\(\cos \theta\)[/tex]
The cosine of angle [tex]\(\theta\)[/tex] is given by the ratio of the adjacent side (x-coordinate) to the hypotenuse.
[tex]\[
\cos \theta = \frac{x}{r}
\][/tex]
Substituting the values:
[tex]\[
\cos \theta = \frac{-3}{\sqrt{73}} \approx -0.351
\][/tex]
3. Calculate [tex]\(\sec \theta\)[/tex]
The secant of angle [tex]\(\theta\)[/tex] is the reciprocal of the cosine function.
[tex]\[
\sec \theta = \frac{1}{\cos \theta}
\][/tex]
Using the value of [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sec \theta \approx \frac{1}{-0.351} \approx -2.848
\][/tex]
4. Calculate [tex]\(\cot \theta\)[/tex]
The cotangent of angle [tex]\(\theta\)[/tex] is given by the ratio of the adjacent side to the opposite side (y-coordinate).
[tex]\[
\cot \theta = \frac{x}{y}
\][/tex]
Substituting the given values:
[tex]\[
\cot \theta = \frac{-3}{8} = -0.375
\][/tex]
Therefore, the exact values are:
[tex]\[
\cos \theta \approx -0.351
\][/tex]
[tex]\[
\sec \theta \approx -2.848
\][/tex]
[tex]\[
\cot \theta = -0.375
\][/tex]
These are the trigonometric values for the given point [tex]\((-3, 8)\)[/tex] on the terminal side of [tex]\(\theta\)[/tex].
