Respuesta :
Answer: [tex]-\frac{169}{120}[/tex]
I'm assuming you mean this:
[tex]cosec(2sin^{-1}(\frac{-12}{13}))[/tex]
Recall what the cosecant function represents. It is the reciprocal of a sine function, and is often written in this form:
[tex]\frac{1}{sinx}[/tex]
So, using the sine relationship:
[tex]cosec(2sin^{-1}(\frac{-12}{13})) = \frac{1}{sin(2sin^{-1}(\frac{-12}{13}))}[/tex]
Let [tex]u = sin^{-1}(\frac{-12}{13})[/tex]
[tex]cosec(2sin^{-1}(\frac{-12}{13})) = \frac{1}{sin(2u)}[/tex]
[tex]= \frac{1}{2sinu \cdot cosu}[/tex]
[tex]= \frac{1}{2sin(sin^{-1}(\frac{-12}{13})) \cdot cos(sin^{-1}(\frac{-12}{13}))}[/tex]
[tex]= \frac{1}{\frac{-24}{13} \cdot \sqrt{1 - (\frac{-12}{13})^{2}}}[/tex]
[tex]= \frac{1}{\frac{-24}{13}} \cdot \frac{1}{\sqrt{1 - \frac{144}{169}}}[/tex]
[tex]= -\frac{13}{24} \cdot \frac{1}{\sqrt{\frac{25}{169}}}[/tex]
[tex]= -\frac{13}{24} \cdot \frac{1}{\frac{5}{13}}[/tex]
[tex]= -\frac{13}{24} \cdot \frac{13}{5}[/tex]
[tex]= -\frac{169}{120}[/tex]
I'm assuming you mean this:
[tex]cosec(2sin^{-1}(\frac{-12}{13}))[/tex]
Recall what the cosecant function represents. It is the reciprocal of a sine function, and is often written in this form:
[tex]\frac{1}{sinx}[/tex]
So, using the sine relationship:
[tex]cosec(2sin^{-1}(\frac{-12}{13})) = \frac{1}{sin(2sin^{-1}(\frac{-12}{13}))}[/tex]
Let [tex]u = sin^{-1}(\frac{-12}{13})[/tex]
[tex]cosec(2sin^{-1}(\frac{-12}{13})) = \frac{1}{sin(2u)}[/tex]
[tex]= \frac{1}{2sinu \cdot cosu}[/tex]
[tex]= \frac{1}{2sin(sin^{-1}(\frac{-12}{13})) \cdot cos(sin^{-1}(\frac{-12}{13}))}[/tex]
[tex]= \frac{1}{\frac{-24}{13} \cdot \sqrt{1 - (\frac{-12}{13})^{2}}}[/tex]
[tex]= \frac{1}{\frac{-24}{13}} \cdot \frac{1}{\sqrt{1 - \frac{144}{169}}}[/tex]
[tex]= -\frac{13}{24} \cdot \frac{1}{\sqrt{\frac{25}{169}}}[/tex]
[tex]= -\frac{13}{24} \cdot \frac{1}{\frac{5}{13}}[/tex]
[tex]= -\frac{13}{24} \cdot \frac{13}{5}[/tex]
[tex]= -\frac{169}{120}[/tex]
