Problem 18 - Precalculus Essentials

Question

Find the exact value of each expression. $$\cos \left(240^{\circ}+45^{\circ}\right)$$

Step-by-Step Solution

Answer

\(\cos(240^{\circ}+45^{\circ}) = -((- \sqrt{2} + \sqrt{6})/ 4\)

Step 1: Conversion

First, convert the sum of angles expressed in degrees into a single angle, by addition. Therefore, \(240^{\circ}+45^{\circ}= 285^{\circ}\)

Step 2: Reference Angle

Identify the equivalent reference angle in the first quadrant, knowing the angle is in the third quadrant. So the reference angle is \(285^{\circ}-180^{\circ}= 105^{\circ}\)

Step 3: Calculate Cosine

Find the cosine of the reference angle, knowing that cosine of an angle in the third quadrant is negative. Hence, \(\cos(285^{\circ}) = -\cos(105^{\circ})\)

Step 4: Breakdown of Angle

Breakdown the \(105^{\circ}\) into sum of two known angles: \(105^{\circ}= 60^{\circ}+ 45^{\circ}\)

Step 5: Use Cosine of Sum of Angles Rule

Apply the rule of cosine of sum of two angles on \(60^{\circ}+ 45^{\circ}\), which is \(\cos(A + B) = \cos A \cos B - \sin A \sin B\). So \(\cos(105^{\circ}) = -(\cos(60^{\circ})\cos(45^{\circ}) - \sin(60^{\circ})\sin(45^{\circ}))\)

Step 6: Calculation

Perform the calculation, substituting known values from special angles \(\cos(60^{\circ}) = 1/2\), \(\cos(45^{\circ}) = \sin(45^{\circ}) = \sin(60^{\circ}) = \sqrt{2}/2\). Hence, \(\cos(285^{\circ}) = -((1/2) * (\sqrt{2}/2) - (\sqrt{2}/2) * (\sqrt{3}/2) ) = -((- \sqrt{2} + \sqrt{6})/ 4\)