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Chapter 5: Problem 37

Find the exact value of the expression. $$\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ}$$

Short Answer

Expert verified
The simplified and exact value of the expression \(\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ}\) is \(\sqrt{3}/2\).

Step by step solution

01

Identify the Formula

Notice that the given expression \(\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ}\) is similar to the sine difference identity \(\sin(A - B) = \sin A \cos B - \cos A \sin B\). Here, \(A = 120^{\circ}\) and \(B = 60^{\circ}\).
02

Apply the Formula

Use the identified formula to rewrite the expression. From here, the exercise becomes easier: \(\sin 120^{\circ} \cos 60^{\circ}-\cos 120^{\circ} \sin 60^{\circ} = \sin(A - B)= \sin(120 - 60)=\sin 60^{\circ}\)
03

Evaluate the Expression

Now evaluate \(\sin 60^{\circ}\). From the unit circle, \(\sin 60^{\circ} = \sqrt{3}/2\). So, the original expression simplifies to \(\sqrt{3}/2\).

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