Answer :
To find the angle θ in degrees for the point [tex]\((-5\sqrt{3}, 15)\)[/tex] that lies on the terminal side of an angle in standard position, follow these steps:
### Step-by-Step Solution:
1. Identify the Coordinates:
- The coordinates of the point are [tex]\((-5\sqrt{3}, 15)\)[/tex].
- [tex]\( x = -5\sqrt{3} \)[/tex]
- [tex]\( y = 15 \)[/tex]
2. Recall the Definition of the Arctangent Function:
- The arctangent function, [tex]\(\arctan(y/x)\)[/tex], gives the angle whose tangent is [tex]\(y/x\)[/tex].
3. Calculate the Ratio [tex]\(y/x\)[/tex]:
- [tex]\[ \frac{y}{x} = \frac{15}{-5\sqrt{3}} = \frac{15}{-5 \cdot \sqrt{3}} = \frac{15}{-5} \cdot \frac{1}{\sqrt{3}} = -3 \cdot \frac{1}{\sqrt{3}} = -\frac{3}{\sqrt{3}} = -\sqrt{3} \][/tex]
4. Calculate the Reference Angle:
- The reference angle whose tangent is [tex]\(-\sqrt{3}\)[/tex] corresponds to 60 degrees (because [tex]\(\tan(60^\circ) = \sqrt{3}\)[/tex]).
5. Determine the Correct Quadrant:
- Since the coordinate [tex]\((-5\sqrt{3}, 15)\)[/tex] places us in the second quadrant (where x is negative and y is positive), the angle θ measured counterclockwise from the positive x-axis to this point must be [tex]\(180^\circ - 60^\circ\)[/tex].
6. Calculate the Angle:
- [tex]\[ \theta = 180^\circ - 60^\circ = 120^\circ \][/tex]
### Conclusion:
The value of the angle θ in degrees is:
[tex]\[ \boxed{120^\circ} \][/tex]
### Step-by-Step Solution:
1. Identify the Coordinates:
- The coordinates of the point are [tex]\((-5\sqrt{3}, 15)\)[/tex].
- [tex]\( x = -5\sqrt{3} \)[/tex]
- [tex]\( y = 15 \)[/tex]
2. Recall the Definition of the Arctangent Function:
- The arctangent function, [tex]\(\arctan(y/x)\)[/tex], gives the angle whose tangent is [tex]\(y/x\)[/tex].
3. Calculate the Ratio [tex]\(y/x\)[/tex]:
- [tex]\[ \frac{y}{x} = \frac{15}{-5\sqrt{3}} = \frac{15}{-5 \cdot \sqrt{3}} = \frac{15}{-5} \cdot \frac{1}{\sqrt{3}} = -3 \cdot \frac{1}{\sqrt{3}} = -\frac{3}{\sqrt{3}} = -\sqrt{3} \][/tex]
4. Calculate the Reference Angle:
- The reference angle whose tangent is [tex]\(-\sqrt{3}\)[/tex] corresponds to 60 degrees (because [tex]\(\tan(60^\circ) = \sqrt{3}\)[/tex]).
5. Determine the Correct Quadrant:
- Since the coordinate [tex]\((-5\sqrt{3}, 15)\)[/tex] places us in the second quadrant (where x is negative and y is positive), the angle θ measured counterclockwise from the positive x-axis to this point must be [tex]\(180^\circ - 60^\circ\)[/tex].
6. Calculate the Angle:
- [tex]\[ \theta = 180^\circ - 60^\circ = 120^\circ \][/tex]
### Conclusion:
The value of the angle θ in degrees is:
[tex]\[ \boxed{120^\circ} \][/tex]
