Answer :
To find the coordinates \((x, y)\) on the terminal ray of the angle \(\theta\), we need to use the given trigonometric functions and relate them to the coordinates on the unit circle. Recall that on the unit circle:
- The \(x\)-coordinate is given by \(\cos \theta\)
- The \(y\)-coordinate is given by \(\sin \theta\)
The given trigonometric values for the angle are:
[tex]\[ \sin \theta = -\frac{77}{85}, \quad \cos \theta = \frac{36}{85}, \quad \tan \theta = -\frac{77}{36} \][/tex]
We can translate these trigonometric values into coordinates by considering them relative to their hypotenuse. Here, the hypotenuse can be determined from the Pythagorean identity, although in this instance the values provided already allow us to determine the correct form.
Firstly, we observe that the coordinates are proportional to the given trigonometric ratios, scaled by their hypotenuse. Given the trigonometric values provided:
[tex]\[ \sin \theta = -\frac{77}{85} \quad \text{and} \quad \cos \theta = \frac{36}{85} \][/tex]
These ratios imply the following:
- The \(y\)-coordinate is \(-\frac{77}{85} \times 85 = -77\)
- The \(x\)-coordinate is \(\frac{36}{85} \times 85 = 36\)
So, the corresponding point \((x, y)\) on the terminal side of the angle \(\theta\) considering the original ratios without simplification would be:
[tex]\[ (x, y) = (36, -77) \][/tex]
Checking against the given options, the coordinates match exactly with one of the provided points. Thus, the point that represents the coordinates \((x, y)\) on the terminal ray of the angle \(\theta\) is:
\[
\boxed{(36, -77)}
- The \(x\)-coordinate is given by \(\cos \theta\)
- The \(y\)-coordinate is given by \(\sin \theta\)
The given trigonometric values for the angle are:
[tex]\[ \sin \theta = -\frac{77}{85}, \quad \cos \theta = \frac{36}{85}, \quad \tan \theta = -\frac{77}{36} \][/tex]
We can translate these trigonometric values into coordinates by considering them relative to their hypotenuse. Here, the hypotenuse can be determined from the Pythagorean identity, although in this instance the values provided already allow us to determine the correct form.
Firstly, we observe that the coordinates are proportional to the given trigonometric ratios, scaled by their hypotenuse. Given the trigonometric values provided:
[tex]\[ \sin \theta = -\frac{77}{85} \quad \text{and} \quad \cos \theta = \frac{36}{85} \][/tex]
These ratios imply the following:
- The \(y\)-coordinate is \(-\frac{77}{85} \times 85 = -77\)
- The \(x\)-coordinate is \(\frac{36}{85} \times 85 = 36\)
So, the corresponding point \((x, y)\) on the terminal side of the angle \(\theta\) considering the original ratios without simplification would be:
[tex]\[ (x, y) = (36, -77) \][/tex]
Checking against the given options, the coordinates match exactly with one of the provided points. Thus, the point that represents the coordinates \((x, y)\) on the terminal ray of the angle \(\theta\) is:
\[
\boxed{(36, -77)}
