To find all six trigonometric functions of [tex]\(\theta\)[/tex], given that the point [tex]\(P (8, 15)\)[/tex] is on the terminal side of [tex]\(\theta\)[/tex], we start by identifying the coordinates of [tex]\(P\)[/tex] as [tex]\( (x, y) = (8, 15) \)[/tex].
We can use the Pythagorean theorem to find the hypotenuse [tex]\(r\)[/tex]:
[tex]\[
r = \sqrt{x^2 + y^2} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17
\][/tex]
With [tex]\(r\)[/tex] known, we can now calculate each of the six trigonometric functions:
1. Sine:
[tex]\[
\sin \theta = \frac{y}{r} = \frac{15}{17}
\][/tex]
2. Cosine:
[tex]\[
\cos \theta = \frac{x}{r} = \frac{8}{17}
\][/tex]
3. Tangent:
[tex]\[
\tan \theta = \frac{y}{x} = \frac{15}{8}
\][/tex]
4. Cotangent:
[tex]\[
\cot \theta = \frac{1}{\tan \theta} = \frac{1}{\frac{15}{8}} = \frac{8}{15}
\][/tex]
5. Secant:
[tex]\[
\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{8}{17}} = \frac{17}{8}
\][/tex]
6. Cosecant:
[tex]\[
\csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{15}{17}} = \frac{17}{15}
\][/tex]
Summarizing the trigonometric functions of [tex]\(\theta\)[/tex]:
[tex]\[
\begin{aligned}
\sin \theta &= \frac{15}{17} \\
\cos \theta &= \frac{8}{17} \\
\tan \theta &= \frac{15}{8} \\
\cot \theta &= \frac{8}{15} \\
\sec \theta &= \frac{17}{8} \\
\csc \theta &= \frac{17}{15}
\end{aligned}
\][/tex]
