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Chapter 5: Problem 10

Find the exact value of each expression. (a) \(\cos \left(120^{\circ}+45^{\circ}\right)\) (b) \(\cos 120^{\circ}+\cos 45^{\circ}\)

Short Answer

Expert verified
The exact values of the expressions are: (a) \(\cos(120^{\circ}+45^{\circ}) = -\sqrt{2}/4 - \sqrt{6}/4\) (b) \(\cos 120^{\circ}+\cos 45^{\circ} = \sqrt{2} - 1/2\)

Step by step solution

01

Solve \(\cos (120^{\circ}+45^{\circ})\)

We use the formula for \(\cos(a + b) = \cos a \cdot \cos b - \sin a \cdot \sin b\). In our case, \(a = 120^{\circ}\) and \(b = 45^{\circ}\). This gives us: \(\cos(120^{\circ} + 45^{\circ}) = \cos 120^{\circ} \cdot \cos 45^{\circ} - \sin 120^{\circ} \cdot \sin 45^{\circ}\). We can now replace each term in the expression with their corresponding values: \(\cos 120^{\circ} = -1/2\), \(\cos 45^{\circ} = \sqrt{2}/2\), \(\sin 120^{\circ} = \sqrt{3}/2\), \(\sin 45^{\circ} = \sqrt{2}/2\). Substituting these values gives us: \(-1/2 \cdot \sqrt{2}/2 - \sqrt{3}/2 \cdot \sqrt{2}/2 = -\sqrt{2}/4 - \sqrt{6}/4\). Collecting the terms together gives us \(-\sqrt{2} - \sqrt{6}\).
02

Solve \(\cos 120^{\circ}+\cos 45^{\circ}\)

Here we just have to calculate the sum of the cosine functions. We know that \(\cos 120^{\circ} = -1/2\) and \(\cos 45^{\circ} = \sqrt{2}/2\). Adding these values together gives us: \(-1/2 + \sqrt{2}/2 = \sqrt{2} - 1/2\).

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