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Three trigonometric functions for a given angle are shown below:

[tex]\[
\csc \theta = \frac{13}{12}, \sec \theta = -\frac{13}{5}, \cot \theta = -\frac{5}{12}
\][/tex]

What are the coordinates of point [tex]\((x, y)\)[/tex] on the terminal ray of angle [tex]\(\theta\)[/tex], assuming that the values above were not simplified?

A. [tex]\((-5, 12)\)[/tex]
B. [tex]\((5, -12)\)[/tex]
C. [tex]\((-12, 5)\)[/tex]
D. [tex]\((12, -5)\)[/tex]

Sagot :

GinnyAnswer
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]
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