Problem 17 - Precalculus Essentials

Question

Find the exact value of each expression. $$\cos \left(135^{\circ}+30^{\circ}\right)$$

Step-by-Step Solution

Answer

The exact value of \(\cos \left(135^{\circ}+30^{\circ}\right)\) is \(-\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\).

Step 1: Convert Degrees to Radians

Initially, we must convert the angles from degrees to radians, as trigonometric functions usually operate in radians. Therefore, \(135^{\circ}\) and \(30^{\circ}\) should be converted to \(\frac{3\pi}{4}\) and \(\frac{\pi}{6}\) respectively.

Step 2: Apply the Cosine of Sum of Angles Formula

Now we can apply the cosine of the sum of two angles formula, i.e., \(\cos(a+b) = \cos a \cos b - \sin a \sin b\). Substituting \(a = \frac{3\pi}{4}\) and \(b = \frac{\pi}{6}\) into the formula, we end up with \(\cos \left(\frac{3\pi}{4} + \frac{\pi}{6}\right) = \cos \frac{3\pi}{4} \cos \frac{\pi}{6} - \sin \frac{3\pi}{4} \sin \frac{\pi}{6}\).

Step 3: Calculate the Individual Trigonometric Functions

Now solve the individual cosine and sine functions. From the unit circle or standard trigonometric values, we know that \(\cos \frac{3\pi}{4} = -\frac{\sqrt{2}}{2}\), \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\), \(\sin \frac{3\pi}{4} = \frac{\sqrt{2}}{2}\), and \(\sin \frac{\pi}{6} = \frac{1}{2}\). Substitute these values into the equation from Step 2.

Step 4: Complete the Calculation

After substituting the values, we end up with \(-\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2}\). Simplify to get \(-\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}\).