Calculate the sine and cosine values for $225^{circ}$ using the unit circle.
Answer 1
Thomas Walker
The unit circle helps us determine the sine and cosine values for any given angle. For the angle $225^{\circ}$, we need to find its location on the unit circle.
The angle $225^{\circ}$ is in the third quadrant. In this quadrant, both sine and cosine values are negative. We can also express $225^{\circ}$ as $180^{\circ} + 45^{\circ}$, where $45^{\circ}$ is a reference angle.
From the unit circle, we know that the coordinates for $45^{\circ}$ are $(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$. Since $225^{\circ}$ is in the third quadrant, the sine and cosine values will be negative:
$\cos(225^{\circ}) = -\frac{\sqrt{2}}{2}$
$\sin(225^{\circ}) = -\frac{\sqrt{2}}{2}$
Thus, the sine and cosine values for $225^{\circ}$ are:
$\cos(225^{\circ}) = -\frac{\sqrt{2}}{2}$
$\sin(225^{\circ}) = -\frac{\sqrt{2}}{2}$
Answer 2
James Taylor
To find the sine and cosine values for $225^{circ}$ on the unit circle, we first note that $225^{circ}$ is located in the third quadrant. In this quadrant, the sine and cosine values are both negative.
The angle $225^{circ}$ can be represented as $180^{circ} + 45^{circ}$. The reference angle here is $45^{circ}$, whose coordinates on the unit circle are $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$.
Since $225^{circ}$ is in the third quadrant, we take the negative of these coordinates:
$cos(225^{circ}) = -frac{sqrt{2}}{2}$
$sin(225^{circ}) = -frac{sqrt{2}}{2}$
Thus, the values for $225^{circ}$ are:
$cos(225^{circ}) = -frac{sqrt{2}}{2}$
$sin(225^{circ}) = -frac{sqrt{2}}{2}$
Answer 3
Olivia Lee
First, locate $225^{circ}$ on the unit circle. It is in the third quadrant, where both sine and cosine are negative.
The angle $225^{circ}$ can be written as $180^{circ} + 45^{circ}$. The reference angle $45^{circ}$ has coordinates $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$.
Therefore,
$cos(225^{circ}) = -frac{sqrt{2}}{2}$
$sin(225^{circ}) = -frac{sqrt{2}}{2}$
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