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]
