To determine the coordinates [tex]\((x, y)\)[/tex] on the unit circle for the angle [tex]\(315^\circ\)[/tex], we need to follow these steps:
1. Convert the Angle from Degrees to Radians:
The angle we have is [tex]\(315^\circ\)[/tex].
[tex]\[
\text{Angle in radians} = \theta = 315^\circ \times \left(\frac{\pi}{180^\circ}\right)
\][/tex]
2. Identify Coordinates Using Trigonometric Functions:
On the unit circle, the coordinates [tex]\((x, y)\)[/tex] corresponding to an angle [tex]\(\theta\)[/tex] in radians are given by:
[tex]\[
x = \cos(\theta)
\][/tex]
[tex]\[
y = \sin(\theta)
\][/tex]
3. Calculate [tex]\(x\)[/tex] and [tex]\(y\)[/tex]:
- For [tex]\(x\)[/tex]:
[tex]\[
x = \cos(315^\circ)
\][/tex]
- For [tex]\(y\)[/tex]:
[tex]\[
y = \sin(315^\circ)
\][/tex]
4. Determine the Exact Values:
- The cosine and sine of [tex]\(315^\circ\)[/tex]:
[tex]\[
x = \cos(315^\circ) = \cos\left(360^\circ - 45^\circ\right) = \cos(360^\circ) \cos(45^\circ) + \sin(360^\circ) \sin(45^\circ)
\][/tex]
Since [tex]\( \cos(360^\circ) = 1 \)[/tex] and [tex]\( \sin(360^\circ) = 0 \)[/tex]:
[tex]\[
x = 1 \cdot \cos(45^\circ) + 0 \cdot \sin(45^\circ) = \cos(45^\circ)
\][/tex]
[tex]\[
\cos(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
- Similarly, for [tex]\(y\)[/tex]:
[tex]\[
y = \sin(315^\circ) = \sin(360^\circ - 45^\circ) = \sin(360^\circ) \cos(45^\circ) - \cos(360^\circ) \sin(45^\circ)
\][/tex]
Since [tex]\( \sin(360^\circ) = 0 \)[/tex] and [tex]\( \cos(360^\circ) = 1 \)[/tex]:
[tex]\[
y = 0 \cdot \cos(45^\circ) - 1 \cdot \sin(45^\circ) = -\sin(45^\circ)
\][/tex]
[tex]\[
\sin(45^\circ) = \frac{\sqrt{2}}{2}
\][/tex]
5. Combine Results:
Therefore:
[tex]\[
x = \frac{\sqrt{2}}{2}
\][/tex]
[tex]\[
y = -\frac{\sqrt{2}}{2}
\][/tex]
The point [tex]\((x, y)\)[/tex] on the unit circle corresponding to the angle [tex]\(315^\circ\)[/tex] is:
[tex]\[
\left(0.7071067811865474, -0.7071067811865477\right)
\][/tex]
