Question

Express as a trigonometric function of one angle. a) \cos2\sin(-9)-\cos9\sin2 Find the exact value of the expression. b) \sin\left(\arcsin\frac{\sqrt{3}}{2}+\arccos0\right)

Express as a trigonometric function of one angle.
a) \(\cos2\sin(-9)-\cos9\sin2\)
Find the exact value of the expression.
b) \(\sin\left(\arcsin\frac{\sqrt{3}}{2}+\arccos0\right)\)

Answers (1)

2021-06-08

Step 1
\(\cos2\sin(-9)-\cos9\sin2=\cos2-\sin q-\cos9\sin2\)
\(\sin(-\theta)=-\sin\theta\)
\(=-\cos2\sin9-\cos9\sin2\)
\(=-(\sin9\cos2+\cos9\sin2)\)
\(=-\sin(9+2)\)
\(=-\sin11\)
\(\because\sin(A+B)\)
\(=\sin A\cos B+\cos A\sin B\)
\(=\sin(-11)\)
\(\because\sin(-\theta)=-\sin\theta\)
Step 2
\(\sin[ \arcsin\frac{\sqrt{3}}{2}+\arccos0]=\sin[\sin^{-1}\left(\frac{\sqrt{3}}{2}\right)+\cos^{-1}(0)]\)
\(=\sin[60^{\circ}+90^{\circ}]=\sin(150^{\circ})\)
\(=\sin(90^{\circ}+60^{\circ})\)
\(\because\sin60^{\circ}=\frac{\sqrt{3}}{2}\)
\(=\cos60^{\circ}\)
\(\cos90^{\circ}=0\)
\(=\frac{1}{2}\)
\(\cos60^{\circ}=\frac{1}{2}\)

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