Let's analyze the given point [tex]\((-5, -7)\)[/tex] which lies on the terminal side of angle [tex]\(\theta\)[/tex].
1. Calculate the hypotenuse [tex]\( r \)[/tex]:
The hypotenuse [tex]\( r \)[/tex] can be found using the Pythagorean Theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
With [tex]\( x = -5 \)[/tex] and [tex]\( y = -7 \)[/tex]:
[tex]\[
r = \sqrt{(-5)^2 + (-7)^2} = \sqrt{25 + 49} = \sqrt{74}
\][/tex]
2. Calculate [tex]\( \cos \theta \)[/tex]:
The cosine of angle [tex]\(\theta\)[/tex] is given by:
[tex]\[
\cos \theta = \frac{x}{r}
\][/tex]
Plugging in [tex]\( x = -5 \)[/tex] and [tex]\( r = \sqrt{74} \)[/tex]:
[tex]\[
\cos \theta = \frac{-5}{\sqrt{74}}
\][/tex]
Simplify if necessary, but we already see:
[tex]\[
\cos \theta \approx -0.5812381937190965
\][/tex]
3. Calculate [tex]\( \csc \theta \)[/tex]:
The cosecant [tex]\( \csc \theta \)[/tex] is the reciprocal of the sine [tex]\( \sin \theta \)[/tex]:
[tex]\[
\csc \theta = \frac{r}{y}
\][/tex]
We know [tex]\( r = \sqrt{74} \)[/tex] and [tex]\( y = -7 \)[/tex]:
[tex]\[
\csc \theta = \frac{\sqrt{74}}{-7}
\][/tex]
Again, simplify:
[tex]\[
\csc \theta \approx -1.228903609577518
\][/tex]
4. Calculate [tex]\( \tan \theta \)[/tex]:
The tangent of angle [tex]\(\theta\)[/tex] is given by:
[tex]\[
\tan \theta = \frac{y}{x}
\][/tex]
With [tex]\( y = -7 \)[/tex] and [tex]\( x = -5 \)[/tex]:
[tex]\[
\tan \theta = \frac{-7}{-5} = \frac{7}{5}
\][/tex]
Simplifying gives us:
[tex]\[
\tan \theta = 1.4
\][/tex]
So the final values are:
- [tex]\( \cos \theta \approx -0.5812381937190965 \)[/tex]
- [tex]\( \csc \theta \approx -1.228903609577518 \)[/tex]
- [tex]\( \tan \theta = 1.4 \)[/tex]
