91Ó°ÊÓ

First, we need to find the reference angle for the given angle, -105 degrees. The reference angle is the angle formed by the terminal side of the given angle and the x-axis. It is always positive and is always less than 180 degrees.To find it, we calculate \(105 \mod 180 = 75\). So the reference angle is \(75^{\circ}\)

For standard degrees like 30, 45, 60, and their multiples, we use special triangles to determine their sine, cosine, and tangent values. Unfortunately, angle 75 degrees is not a standard angle. However, it can be expressed as the sum \(75^\circ = 45^\circ + 30^\circ\). Then, the exact values for 75 degrees can be determined with the help of addition formulas for sine and cosine: \(\sin(x + y) = \sin(x) \cos(y) + \cos(x) \sin(y)\) and \(\cos(x + y) = \cos(x) \cos(y) - \sin(x) \sin(y)\). This gives \(\sin(75^\circ) = \sin(45^\circ) \cos(30^\circ) + \cos(45^\circ) \sin(30^\circ) = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times 0.5 = \frac{\sqrt{6} + \sqrt{2}}{4}\) and \(\cos(75^\circ) = \cos(45^\circ) \cos(30^\circ) - \sin(45^\circ) \sin(30^\circ) = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \times 0.5 = \frac{\sqrt{6} - \sqrt{2}}{4}\). Lastly, for tangent, the formula \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) gives \(\tan(75^\circ) = \frac{\sin(75^\circ)}{\cos(75^\circ)} = \frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\)

The original angle -105 degrees is in the fourth quadrant. In this quadrant, sine is negative, cosine is positive, and tangent is negative. So, for \(-105^\circ\), \(\sin(-105^\circ) = -\sin(75^\circ) = -\frac{\sqrt{6} + \sqrt{2}}{4}\), \(\cos(-105^\circ) = \cos(75^\circ) = \frac{\sqrt{6} - \sqrt{2}}{4}\), and \(\tan(-105^\circ) = -\tan(75^\circ) = -\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\)