Given the point [tex]\((-6, -5)\)[/tex] on the terminal side of angle [tex]\(\theta\)[/tex], we need to find the exact values of [tex]\(\sin \theta\)[/tex], [tex]\(\sec \theta\)[/tex], and [tex]\(\tan \theta\)[/tex].
### Step-by-Step Solution:
1. Calculate the hypotenuse [tex]\( r \)[/tex]:
The coordinates of the point are [tex]\((x, y) = (-6, -5)\)[/tex]. The hypotenuse [tex]\( r \)[/tex] (the distance from the origin to this point) can be found using the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Plugging in the values [tex]\( x = -6 \)[/tex] and [tex]\( y = -5 \)[/tex]:
[tex]\[
r = \sqrt{(-6)^2 + (-5)^2} = \sqrt{36 + 25} = \sqrt{61} \approx 7.810249675906654
\][/tex]
2. Calculate [tex]\(\sin \theta\)[/tex]:
Sine is defined as:
[tex]\[
\sin \theta = \frac{y}{r}
\][/tex]
Substituting [tex]\( y = -5 \)[/tex] and [tex]\( r \approx 7.810249675906654 \)[/tex]:
[tex]\[
\sin \theta = \frac{-5}{7.810249675906654} \approx -0.6401843996644799
\][/tex]
3. Calculate [tex]\(\sec \theta\)[/tex]:
Secant is the reciprocal of cosine. Cosine is defined as:
[tex]\[
\cos \theta = \frac{x}{r}
\][/tex]
And so, secant is defined as:
[tex]\[
\sec \theta = \frac{r}{x}
\][/tex]
Substituting [tex]\( x = -6 \)[/tex] and [tex]\( r \approx 7.810249675906654 \)[/tex]:
[tex]\[
\sec \theta = \frac{7.810249675906654}{-6} \approx -1.3017082793177757
\][/tex]
4. Calculate [tex]\(\tan \theta\)[/tex]:
Tangent is defined as:
[tex]\[
\tan \theta = \frac{y}{x}
\][/tex]
Substituting [tex]\( y = -5 \)[/tex] and [tex]\( x = -6 \)[/tex]:
[tex]\[
\tan \theta = \frac{-5}{-6} = \frac{5}{6} \approx 0.8333333333333334
\][/tex]
### Summary:
- [tex]\(\sin \theta \approx -0.6401843996644799\)[/tex]
- [tex]\(\sec \theta \approx -1.3017082793177757\)[/tex]
- [tex]\(\tan \theta \approx 0.8333333333333334\)[/tex]
These results give us the exact values of the trigonometric functions for the given angle [tex]\(\theta\)[/tex].
