To find the coordinates of the point [tex]\((x, y)\)[/tex] on the terminal ray of angle [tex]\(\theta\)[/tex], we need to use the given trigonometric functions and their values. The given values are:
[tex]\[ \csc \theta = \frac{13}{12} \][/tex]
[tex]\[ \sec \theta = -\frac{13}{5} \][/tex]
[tex]\[ \cot \theta = -\frac{5}{12} \][/tex]
Let's interpret each of these values step-by-step.
1. Cosecant ([tex]\(\csc \theta\)[/tex]):
[tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex]
Since [tex]\(\csc \theta = \frac{13}{12}\)[/tex], we get:
[tex]\[\sin \theta = \frac{12}{13}\][/tex]
This tells us the ratio of the opposite side to the hypotenuse in the right triangle.
2. Secant ([tex]\(\sec \theta\)[/tex]):
[tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex]
Since [tex]\(\sec \theta = -\frac{13}{5}\)[/tex], we get:
[tex]\[\cos \theta = -\frac{5}{13}\][/tex]
This tells us the ratio of the adjacent side to the hypotenuse in the right triangle.
3. Cotangent ([tex]\(\cot \theta\)[/tex]):
[tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]
Since [tex]\(\cot \theta = -\frac{5}{12}\)[/tex], it confirms:
[tex]\[\cot \theta = -\frac{\cos \theta}{\sin \theta} = \frac{-5}{12}\][/tex]
Now, we need to identify the lengths of the sides of the triangle that allows us to find the coordinates [tex]\( (x, y) \)[/tex]:
- Opposite side: from [tex]\(\sin \theta = \frac{12}{13}\)[/tex], we know the opposite side is [tex]\(12\)[/tex].
- Hypotenuse: from both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex], we know it is [tex]\(13\)[/tex].
- Adjacent side: from [tex]\(\cos \theta = -\frac{5}{13}\)[/tex], we know the adjacent side is [tex]\(-5\)[/tex].
The coordinates are determined by the adjacent side (x) and the opposite side (y) values:
[tex]\[x = \text{adjacent} = -5\][/tex]
[tex]\[y = \text{opposite} = 12\][/tex]
Thus, the coordinates [tex]\((x, y)\)[/tex] are:
[tex]\[
(-5, 12)
\][/tex]
Hence, among the given options, the correct coordinates of the point on the terminal ray of angle [tex]\(\theta\)[/tex] are:
[tex]\[
(-5, 12)
\][/tex]
