To determine the coordinates of a point [tex]\((x, y)\)[/tex] on the terminal ray of a given angle [tex]\(\theta\)[/tex], when given the trigonometric values [tex]\(\sin \theta = -\frac{77}{85}\)[/tex], [tex]\(\cos \theta = \frac{36}{85}\)[/tex], and [tex]\(\tan \theta = -\frac{77}{36}\)[/tex], we perform the following steps:
1. Identify the relationship between the given sine and cosine values and the coordinates:
For any angle [tex]\(\theta\)[/tex], the coordinates [tex]\((x, y)\)[/tex] on the unit circle can be identified by knowing:
- [tex]\(x = r \cos \theta\)[/tex]
- [tex]\(y = r \sin \theta\)[/tex]
Where [tex]\(r\)[/tex] is the radius of the circle. In this case, because these values are given in terms of a simplified form within a right triangle, we can directly consider them as multiples of a hypotenuse [tex]\(r\)[/tex].
2. Assumption of a common radius:
Since [tex]\(\sin \theta = -\frac{77}{85}\)[/tex] and [tex]\(\cos \theta = \frac{36}{85}\)[/tex], it's clear that both sine and cosine are derived with the same hypotenuse value of 85.
3. Express coordinates using the given sine and cosine values:
Hence:
- [tex]\(x = r \cos \theta = 36\)[/tex] (since the original hypotenuse value [tex]\(r\)[/tex] would just cancel out)
- [tex]\(y = r \sin \theta = -77\)[/tex] (similarly here)
4. Conclusions and final coordinates:
Given these calculations:
- The coordinate [tex]\(x\)[/tex] derived from [tex]\(\cos \theta = \frac{36}{85}\)[/tex] is 36.
- The coordinate [tex]\(y\)[/tex] derived from [tex]\(\sin \theta = -\frac{77}{85}\)[/tex] is -77.
Therefore, the coordinates of the point [tex]\((x,y)\)[/tex] on the terminal ray of the angle [tex]\(\theta\)[/tex] are [tex]\((36, -77)\)[/tex].
By matching this result against the given options:
- [tex]\((-77, -36)\)[/tex]
- [tex]\((-77, 36)\)[/tex]
- [tex]\((-36, 77)\)[/tex]
- [tex]\((36, -77)\)[/tex]
We see that the correct coordinates are:
[tex]$
(36, -77)
$[/tex]
