How do you find two exact values of sin((cos−1)(√56))?

1 Answer
sente
Dec 12, 2015

The two exact values of sin(cos−1(√56)) are
√316 and −√316

Explanation:

Let's consider the right triangle with angle x where cos(x)=√56

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Then cos−1(√56)=x and so
sin(cos−1(√56))=sin(x)=y6

By the Pythagorean theorem,

(√5)2+y2=62⇒y=√36−5=√31

So we have one value of sin(cos−1(√56)) as √316

But assuming we are looking at cos−1 as multivalued, looking at it geometrically loses one solution, as we only look at one possibility for cos−1(√56). To get the other, note that as as the cosine function is even,

cos(−x)=cos(x)=√56

So we need to consider −x as our other value. Then, as the sine function is odd,

sin(−x)=−sin(x)=−√316

Thus the two exact values of sin(cos−1(√56)) are √316 and −√316

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