To find the values of the six trigonometric functions for angle [tex]\( B \)[/tex], where the terminal side passes through the point [tex]\((-1, 10)\)[/tex], we follow these steps:
1. Identify the coordinates:
Let [tex]\( x = -1 \)[/tex] and [tex]\( y = 10 \)[/tex].
2. Calculate the hypotenuse (r):
[tex]\[
r = \sqrt{x^2 + y^2} = \sqrt{(-1)^2 + 10^2} = \sqrt{1 + 100} = \sqrt{101} \approx 10.05
\][/tex]
3. Determine the trigonometric functions:
- Sine ([tex]\(\sin \beta\)[/tex]):
[tex]\[
\sin \beta = \frac{y}{r} = \frac{10}{10.05} \approx 1.0
\][/tex]
- Cosine ([tex]\(\cos \beta\)[/tex]):
[tex]\[
\cos \beta = \frac{x}{r} = \frac{-1}{10.05} \approx -0.1
\][/tex]
- Tangent ([tex]\(\tan \beta\)[/tex]):
[tex]\[
\tan \beta = \frac{y}{x} = \frac{10}{-1} = -10.0
\][/tex]
- Cosecant ([tex]\(\csc \beta\)[/tex]):
[tex]\[
\csc \beta = \frac{r}{y} = \frac{10.05}{10} \approx 1.0
\][/tex]
- Secant ([tex]\(\sec \beta\)[/tex]):
[tex]\[
\sec \beta = \frac{r}{x} = \frac{10.05}{-1} \approx -10.0
\][/tex]
- Cotangent ([tex]\(\cot \beta\)[/tex]):
[tex]\[
\cot \beta = \frac{x}{y} = \frac{-1}{10} \approx -0.1
\][/tex]
So, the rounded values of the six trigonometric functions of [tex]\( B \)[/tex] are:
[tex]\[
\begin{aligned}
\sin (\beta) & = 1.0 \\
\cos (\beta) & = -0.1 \\
\tan (\beta) & = -10.0 \\
\csc (\beta) & = 1.0 \\
\sec (\beta) & = -10.0 \\
\cot (\beta) & = -0.1
\end{aligned}
\][/tex]
