To find the coordinates of point [tex]\( (x, y) \)[/tex] on the terminal ray of angle [tex]\( \theta \)[/tex] given the trigonometric functions [tex]\(\csc \theta = \frac{13}{12}\)[/tex], [tex]\(\sec \theta = -\frac{13}{5}\)[/tex], and [tex]\(\cot \theta = -\frac{5}{12}\)[/tex], we can follow this detailed step-by-step process.
### Step 1: Understanding the Trigonometric Functions
The given trigonometric functions are:
1. [tex]\(\csc \theta = \frac{13}{12}\)[/tex]
2. [tex]\(\sec \theta = -\frac{13}{5}\)[/tex]
3. [tex]\(\cot \theta = -\frac{5}{12}\)[/tex]
We know the relationships between these functions and the basic trigonometric functions ([tex]\(\sin\)[/tex], [tex]\(\cos\)[/tex], [tex]\(\tan\)[/tex]):
- [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex]
- [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex]
- [tex]\(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)[/tex]
### Step 2: Find [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]
Using the given values:
- [tex]\(\csc \theta = \frac{13}{12}\)[/tex],
So, [tex]\(\sin \theta = \frac{1}{\csc \theta} = \frac{12}{13}\)[/tex].
- [tex]\(\sec \theta = -\frac{13}{5}\)[/tex],
So, [tex]\(\cos \theta = \frac{1}{\sec \theta} = -\frac{5}{13}\)[/tex].
### Step 3: Determine the Coordinates
The coordinates [tex]\((x, y)\)[/tex] on the terminal ray can be found using the relationships:
- [tex]\( \sin \theta = \frac{y}{r} \)[/tex]
- [tex]\( \cos \theta = \frac{x}{r} \)[/tex]
where [tex]\( r \)[/tex] is the hypotenuse of the right triangle formed by [tex]\((x, y)\)[/tex]. We can see from the values given that [tex]\( r \)[/tex] (the hypotenuse) is 13 because this is consistent in both [tex]\(\csc \theta\)[/tex] and [tex]\(\sec \theta\)[/tex].
However, to confirm, we can do a quick check:
- [tex]\(\sin \theta = \frac{12}{13}\)[/tex] implies [tex]\( y = 12 \)[/tex]
- [tex]\(\cos \theta = -\frac{5}{13}\)[/tex] implies [tex]\( x = -5 \)[/tex]
Thus, the coordinates are:
[tex]\[ (x, y) = (-5, 12) \][/tex]
Based on the given options:
- [tex]\((-5, 12)\)[/tex]
- [tex]\((5, -12)\)[/tex]
- [tex]\((-12, 5)\)[/tex]
- [tex]\((12, -5)\)[/tex]
The correct coordinates are:
[tex]\[ \boxed{(-5, 12)} \][/tex]
