Three trigonometric functions for a given angle are shown below:

[tex]\[ \sin \theta = -\frac{77}{85}, \cos \theta = \frac{36}{85}, \tan \theta = -\frac{77}{36} \][/tex]

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

A. [tex]\((-77, -36)\)[/tex]
B. [tex]\((-77, 36)\)[/tex]
C. [tex]\((-36, 77)\)[/tex]
D. [tex]\((36, -77)\)[/tex]



Answer :

To solve this question, we need to determine the coordinates of the point [tex]\((x, y)\)[/tex] on the terminal ray of angle [tex]\(\theta\)[/tex].

Given the trigonometric values:
[tex]\[ \sin \theta = -\frac{77}{85}, \quad \cos \theta = \frac{36}{85}, \quad \tan \theta = -\frac{77}{36} \][/tex]

The relationship between the trigonometric functions and the coordinates on the unit circle (where the radius [tex]\( r = 1 \)[/tex]) is as follows:
- [tex]\(\sin \theta = \frac{y}{r}\)[/tex]
- [tex]\(\cos \theta = \frac{x}{r}\)[/tex]
- [tex]\(\tan \theta = \frac{y}{x}\)[/tex]

However, in this context, we are not dealing with the unit circle because [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] both have the denominator 85, indicating that [tex]\(r\)[/tex] is actually 85. We need [tex]\(x\)[/tex] and [tex]\(y\)[/tex] proportional to 36 and -77, respectively.

Let's identify [tex]\((x, y)\)[/tex]:
1. According to [tex]\(\cos \theta = \frac{x}{r}\)[/tex], we have:
[tex]\[ \frac{x}{85} = \frac{36}{85} \Rightarrow x = 36 \][/tex]

2. According to [tex]\(\sin \theta = \frac{y}{r}\)[/tex], we have:
[tex]\[ \frac{y}{85} = \frac{-77}{85} \Rightarrow y = -77 \][/tex]

Hence, the coordinates [tex]\((x, y)\)[/tex] that satisfy all given trigonometric conditions are [tex]\((36, -77)\)[/tex].

So, the correct answer is:
[tex]\[ \boxed{(36, -77)} \][/tex]