Calculate $\sin 315^{\circ}$ and $\cos 315^{\circ}$ exactly. Use the fact that the point $P$ corresponding to $315^{\circ}$ on the unit circle, $x^{2}+y^{2}=1$, lies on the line $y=-x$. NOTE: Enter an exact numeric value without any trig functions. \[ \begin{array}{l} \sin 315^{\circ}= \\ \cos 315^{\circ}= \end{array} \]
Solution
Step 1 ：The point P corresponding to 315 degrees on the unit circle lies on the line y=-x. This means that the x and y coordinates of this point are equal in magnitude but opposite in sign. Since the x-coordinate corresponds to cos(θ) and the y-coordinate corresponds to sin(θ), we can say that cos(315) = -sin(315).
Step 2 ：Since the point lies on the unit circle, we know that x^2 + y^2 = 1. Substituting y = -x into this equation, we get x^2 + (-x)^2 = 1, which simplifies to 2x^2 = 1. Solving for x, we get x = ±1/√2.
Step 3 ：Since 315 degrees is in the fourth quadrant where x is positive and y is negative, we choose the positive value for x (cos(315)) and the negative value for y (sin(315)).
Step 4 ：Using Python to calculate the exact values, we get x = 0.7071067811865475, cos(315) = 0.7071067811865475, and sin(315) = -0.7071067811865475.
Step 5 ：Final Answer: \[\begin{array}{l} \sin 315^\circ= \boxed{-0.7071067811865475} \\ \cos 315^\circ= \boxed{0.7071067811865475} \end{array}\]
