Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact values of the sine, cosine, and tangent of the angle. $$-105^{\circ}$$

Step-by-Step Solution

Answer

The exact values for \(-105^{\circ}\) are as follows: \(\sin(-105^{\circ}) = \frac{\sqrt{6} + \sqrt{2}}{4}\), \(\cos(-105^{\circ}) = \frac{-\sqrt{6} + \sqrt{2}}{4}\), \(\tan(-105^{\circ}) = -\frac{\sqrt{6} + \sqrt{2}}{\sqrt{6} - \sqrt{2}}\).

Step 1: Express the Angle in Terms of Known Trigonometric Angles

Given the angle of \(-105^{\circ}\), it can be re-written as \(-90^{\circ}\) and \(-15^{\circ}\), since \(-105^{\circ} = -90^{\circ} - 15^{\circ}\). These are known angles, so their trigonometric values can easily be evaluated.

Step 2: Place the Angle in Correct Quadrant

Negative angles are measured clockwise from the positive x-axis. Therefore, the angle of \(-105^{\circ}\) will lie in the second quadrant. In the second quadrant the sine function is positive, whereas cosine and tangent are negative.

Step 3: Evaluate the Trigonometric Values

The sine of \(-105^{\circ}\) is equivalent to the cosine of \(-15^{\circ}\), which is \(\sqrt{6} + \sqrt{2}\) over \(4\). The cosine, on the other hand, is equivalent to negative of sine of \(-15^{\circ}\), which is \(\sqrt{6} - \sqrt{2}\) over \(4\). The tangent is the sine divided by the cosine, and its value is therefore \(-(\sqrt{6} + \sqrt{2})/(\sqrt{6} - \sqrt{2})\).