To find the exact values of [tex]\(\sin \theta\)[/tex], [tex]\(\sec \theta\)[/tex], and [tex]\(\tan \theta\)[/tex] when given the point [tex]\((-5, -3)\)[/tex] on the terminal side of [tex]\(\theta\)[/tex], follow these steps:
1. Calculate the hypotenuse [tex]\( r \)[/tex]:
The hypotenuse [tex]\( r \)[/tex] is found using the Pythagorean theorem, where [tex]\( r \)[/tex] is the distance from the origin to the point [tex]\((-5, -3)\)[/tex].
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Substituting [tex]\( x = -5 \)[/tex] and [tex]\( y = -3 \)[/tex]:
[tex]\[
r = \sqrt{(-5)^2 + (-3)^2}
\][/tex]
[tex]\[
r = \sqrt{25 + 9}
\][/tex]
[tex]\[
r = \sqrt{34}
\][/tex]
Hence,
[tex]\[
r \approx 5.830951894845301
\][/tex]
2. Determine [tex]\(\sin \theta\)[/tex]:
By definition, [tex]\(\sin \theta\)[/tex] is the ratio of the opposite side over the hypotenuse.
[tex]\[
\sin \theta = \frac{y}{r}
\][/tex]
Substituting [tex]\( y = -3 \)[/tex] and [tex]\( r \approx 5.830951894845301 \)[/tex]:
[tex]\[
\sin \theta = \frac{-3}{5.830951894845301}
\][/tex]
[tex]\[
\sin \theta \approx -0.5144957554275265
\][/tex]
3. Determine [tex]\(\sec \theta\)[/tex]:
By definition, [tex]\(\sec \theta\)[/tex] is the reciprocal of [tex]\(\cos \theta\)[/tex], where [tex]\(\cos \theta\)[/tex] is the ratio of the adjacent side over the hypotenuse.
[tex]\[
\sec \theta = \frac{r}{x}
\][/tex]
Substituting [tex]\( r \approx 5.830951894845301 \)[/tex] and [tex]\( x = -5 \)[/tex]:
[tex]\[
\sec \theta = \frac{5.830951894845301}{-5}
\][/tex]
[tex]\[
\sec \theta \approx -1.1661903789690602
\][/tex]
4. Determine [tex]\(\tan \theta\)[/tex]:
By definition, [tex]\(\tan \theta\)[/tex] is the ratio of the opposite side over the adjacent side.
[tex]\[
\tan \theta = \frac{y}{x}
\][/tex]
Substituting [tex]\( y = -3 \)[/tex] and [tex]\( x = -5 \)[/tex]:
[tex]\[
\tan \theta = \frac{-3}{-5}
\][/tex]
[tex]\[
\tan \theta = 0.6
\][/tex]
Therefore, the values are:
[tex]\[
\sin \theta \approx -0.5144957554275265
\][/tex]
[tex]\[
\sec \theta \approx -1.1661903789690602
\][/tex]
[tex]\[
\tan \theta \approx 0.6
\][/tex]
