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monitta
3 years ago
9

Csc (2sin^-1(-12/13)) that is 2 times inverse sin of -12/13

Mathematics
1 answer:
german3 years ago
3 0
Answer: -\frac{169}{120}

I'm assuming you mean this:

cosec(2sin^{-1}(\frac{-12}{13}))

Recall what the cosecant function represents. It is the reciprocal of a sine function, and is often written in this form:

\frac{1}{sinx}

So, using the sine relationship:

cosec(2sin^{-1}(\frac{-12}{13})) = \frac{1}{sin(2sin^{-1}(\frac{-12}{13}))}

Let u = sin^{-1}(\frac{-12}{13})
cosec(2sin^{-1}(\frac{-12}{13})) = \frac{1}{sin(2u)}
= \frac{1}{2sinu \cdot cosu}
= \frac{1}{2sin(sin^{-1}(\frac{-12}{13})) \cdot cos(sin^{-1}(\frac{-12}{13}))}
= \frac{1}{\frac{-24}{13} \cdot \sqrt{1 - (\frac{-12}{13})^{2}}}
= \frac{1}{\frac{-24}{13}} \cdot \frac{1}{\sqrt{1 - \frac{144}{169}}}
= -\frac{13}{24} \cdot \frac{1}{\sqrt{\frac{25}{169}}}
= -\frac{13}{24} \cdot \frac{1}{\frac{5}{13}}
= -\frac{13}{24} \cdot \frac{13}{5}
= -\frac{169}{120}
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