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home > precalculus ma - 440 cfh - fall 2023 - dr. purl > assessment
assignment 7.2: sum and difference identities
score: 0/100 answered: 0/6
question 1
find the exact value of sin(285°).
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Explanation:

Step1: Rewrite the angle

We can write \(285^{\circ}=240^{\circ} + 45^{\circ}\). The sum - of - angles formula for sine is \(\sin(A + B)=\sin A\cos B+\cos A\sin B\). Here \(A = 240^{\circ}\) and \(B=45^{\circ}\).

Step2: Find the sine and cosine of \(A = 240^{\circ}\)

\(\sin(240^{\circ})=\sin(180^{\circ}+60^{\circ})=-\sin(60^{\circ})=-\frac{\sqrt{3}}{2}\), \(\cos(240^{\circ})=\cos(180^{\circ}+60^{\circ})=-\cos(60^{\circ})=-\frac{1}{2}\).

Step3: Find the sine and cosine of \(B = 45^{\circ}\)

\(\sin(45^{\circ})=\frac{\sqrt{2}}{2}\), \(\cos(45^{\circ})=\frac{\sqrt{2}}{2}\).

Step4: Apply the sum - of - angles formula

\(\sin(285^{\circ})=\sin(240^{\circ}+45^{\circ})=\sin(240^{\circ})\cos(45^{\circ})+\cos(240^{\circ})\sin(45^{\circ})\)

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Answer:

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