Find the sine, cosine, and tangent of each angle using the unit circle.
$\theta=225\degree,\left(-\frac{\sqrt2}{2},-\frac{\sqrt2}{2}\right)$

• A. $\sin\theta=-\frac{\sqrt2}{2},\cos\theta=-\frac{\sqrt2}{2},\tan\theta=2$
• B. $\sin\theta=\frac{\sqrt2}{2},\cos\theta=-\frac{\sqrt2}{2},\tan\theta=-1$
• C. $\sin\theta=-\frac{\sqrt2}{2},\cos\theta=-\frac{\sqrt2}{2},\tan\theta=1$
• D. $\sin\theta=\frac{\sqrt2}{2},\cos\theta=\frac{\sqrt2}{2},\tan\theta=\frac{\sqrt2}{1}$

Answer: (C)

Verified step by step guidance

1. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate plane. Each point on the unit circle corresponds to an angle θ measured from the positive x-axis.
2. For an angle θ = 225°, locate the point on the unit circle. The coordinates of this point are given as (-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}).
3. The x-coordinate of the point on the unit circle represents the cosine of the angle, and the y-coordinate represents the sine of the angle. Therefore, \cos(225°) = -\frac{\sqrt{2}}{2} and \sin(225°) = -\frac{\sqrt{2}}{2}.
4. The tangent of an angle θ is the ratio of the sine to the cosine, \tan(θ) = \frac{\sin(θ)}{\cos(θ)}. For θ = 225°, \tan(225°) = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1.
5. Verify the quadrant: 225° is in the third quadrant where both sine and cosine are negative, confirming that \sin(225°) = -\frac{\sqrt{2}}{2}, \cos(225°) = -\frac{\sqrt{2}}{2}, and \tan(225°) = 1 are correct.