Question

assignment 9.2: sum and difference identities
score: 0/8 answered: 0/8
question 1
find the exact value of \\( \cos ( 165 ^ { \circ } ) \\).
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Explanation:

Step1: Express \(165^{\circ}\) as a sum

\(165^{\circ}=120^{\circ}+45^{\circ}\)

Step2: Use the cosine sum formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\)

Here \(A = 120^{\circ}\), \(B=45^{\circ}\)
\(\cos(120^{\circ}+45^{\circ})=\cos120^{\circ}\cos45^{\circ}-\sin120^{\circ}\sin45^{\circ}\)

Step3: Substitute the values of trigonometric functions

We know that \(\cos120^{\circ}=-\frac{1}{2}\), \(\cos45^{\circ}=\frac{\sqrt{2}}{2}\), \(\sin120^{\circ}=\frac{\sqrt{3}}{2}\), \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\)

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Answer:

\(-\frac{\sqrt{2}+\sqrt{6}}{4}\)