To determine the coordinates [tex]\((x, y)\)[/tex] on the terminal ray of the angle [tex]\(\theta\)[/tex], we need to use the given trigonometric values for [tex]\(\sin \theta\)[/tex], [tex]\(\cos \theta\)[/tex], and [tex]\(\tan \theta\)[/tex]. The values provided are:
[tex]\[
\sin \theta = -\frac{77}{85}, \quad \cos \theta = \frac{36}{85}, \quad \tan \theta = -\frac{77}{36}
\][/tex]
We start by recalling the relationships between these trigonometric ratios and the coordinates on the unit circle or any circle with radius [tex]\(r\)[/tex]. For any angle [tex]\(\theta\)[/tex], these relationships are:
[tex]\[
\sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}
\][/tex]
where [tex]\((x, y)\)[/tex] are the coordinates of the point on the terminal side of the angle [tex]\(\theta\)[/tex], and [tex]\(r\)[/tex] is the hypotenuse, which is the distance from the origin to the point [tex]\((x, y)\)[/tex].
Given [tex]\(\cos \theta = \frac{36}{85}\)[/tex], and [tex]\(\sin \theta = -\frac{77}{85}\)[/tex], and knowing these values correspond to a triangle where [tex]\( r = 85 \)[/tex] (hypotenuse):
Let's solve for the coordinates [tex]\((x, y)\)[/tex]:
- From [tex]\(\cos \theta = \frac{x}{r}\)[/tex] and knowing [tex]\( r = 85 \)[/tex], we have:
[tex]\[
\cos \theta = \frac{x}{85} \implies x = 85 \cdot \frac{36}{85} = 36
\][/tex]
- From [tex]\(\sin \theta = \frac{y}{r}\)[/tex] and knowing [tex]\( r = 85 \)[/tex], we have:
[tex]\[
\sin \theta = \frac{y}{85} \implies y = 85 \cdot -\frac{77}{85} = -77
\][/tex]
Thus, the coordinates [tex]\((x, y)\)[/tex] on the terminal ray of angle [tex]\(\theta\)[/tex] are:
[tex]\[
(36, -77)
\][/tex]
Given the provided options:
- [tex]\((-77, -36)\)[/tex]
- [tex]\((-77, 36)\)[/tex]
- [tex]\((-36, 77)\)[/tex]
- [tex]\((36, -77)\)[/tex]
The correct coordinates are:
[tex]\[
(36, -77)
\][/tex]
