Given the point [tex]\((13, 84)\)[/tex] lies on the terminal side of an angle [tex]\(\alpha\)[/tex] in standard position, we need to find the values of the six trigonometric functions of the angle [tex]\(\alpha\)[/tex].
First, let's calculate the distance [tex]\(r\)[/tex] from the origin to the point (13, 84):
[tex]\[ r = \sqrt{x^2 + y^2} \][/tex]
[tex]\[ r = \sqrt{13^2 + 84^2} \][/tex]
[tex]\[ r = \sqrt{169 + 7056} \][/tex]
[tex]\[ r = \sqrt{7225} \][/tex]
[tex]\[ r = 85 \][/tex]
Now, using [tex]\(x = 13\)[/tex], [tex]\(y = 84\)[/tex], and [tex]\(r = 85\)[/tex], we can find the trigonometric functions:
1. Sine ([tex]\(\sin \alpha\)[/tex]):
[tex]\[ \sin \alpha = \frac{y}{r} \][/tex]
[tex]\[ \sin \alpha = \frac{84}{85} \][/tex]
Thus,
[tex]\[ \sin \alpha = 0.9882352941176471 \][/tex]
2. Cosine ([tex]\(\cos \alpha\)[/tex]):
[tex]\[ \cos \alpha = \frac{x}{r} \][/tex]
[tex]\[ \cos \alpha = \frac{13}{85} \][/tex]
Thus,
[tex]\[ \cos \alpha = 0.15294117647058825 \][/tex]
3. Tangent ([tex]\(\tan \alpha\)[/tex]):
[tex]\[ \tan \alpha = \frac{y}{x} \][/tex]
[tex]\[ \tan \alpha = \frac{84}{13} \][/tex]
Thus,
[tex]\[ \tan \alpha = 6.461538461538462 \][/tex]
4. Cosecant ([tex]\(\csc \alpha\)[/tex]):
[tex]\[ \csc \alpha = \frac{r}{y} \][/tex]
[tex]\[ \csc \alpha = \frac{85}{84} \][/tex]
Thus,
[tex]\[ \csc \alpha = 1.0119047619047619 \][/tex]
5. Secant ([tex]\(\sec \alpha\)[/tex]):
[tex]\[ \sec \alpha = \frac{r}{x} \][/tex]
[tex]\[ \sec \alpha = \frac{85}{13} \][/tex]
Thus,
[tex]\[ \sec \alpha = 6.538461538461538 \][/tex]
6. Cotangent ([tex]\(\cot \alpha\)[/tex]):
[tex]\[ \cot \alpha = \frac{x}{y} \][/tex]
[tex]\[ \cot \alpha = \frac{13}{84} \][/tex]
Thus,
[tex]\[ \cot \alpha = 0.15476190476190477 \][/tex]
Considering these results, the correct selection from the provided options is:
[tex]\[
\sin \alpha = \frac{84}{85}, \cos \alpha = \frac{13}{85}, \tan \alpha = \frac{84}{13}, \csc \alpha = \frac{85}{84}, \sec \alpha = \frac{85}{13}, \cot \alpha = \frac{13}{84}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\sin \alpha = \frac{84}{85}, \cos \alpha = \frac{13}{85}, \tan \alpha = \frac{84}{13}, \csc \alpha = \frac{85}{84}, \sec \alpha = \frac{85}{13}, \cot \alpha = \frac{13}{84}
\][/tex]
