Finding value of sin2theta

[alexedward]

If csctheta = 5, 0 < theta < pi/2, find the value of sin2theta. I'm not sure at all where to start with this problem.

 

[mmm4444bot]

You could start by thinking about the definition of the cosecant function:

1/sin(?)

Set that expression equal to 5, and solve for ? using the arcsine function.

You can approximate ? using a scientific calculator, and then evaluate sin(2?).

Or, if your machine is savvy, you could just ask it to evaluate sin(2*arccsc(5)). :wink:

 

[soroban]

Hello, alexedward!

\(\displaystyle \csc\theta = 5,\;\;0 < \theta < \tfrac{\pi}{2}\)

Find the value of \(\displaystyle \sin2\theta\)

\(\displaystyle \text{We have: }\;\csc\theta \;=\;\frac{5}{1} \;=\;\frac{hyp}{opp}\)

\(\displaystyle \theta\text{ is in a right triangle with: }\pp = 1,\;hyp = 5\)

\(\displaystyle \text{Pythagorus says: }\:adj = \sqrt{24} = 2\sqrt{6}\)

\(\displaystyle \text{Hence: }\:\sin\theta \,=\,\frac{1}{5},\;\;\cos\theta \,=\,\frac{2\sqrt{6}}{5}\;\;{\bf [1]}\)


\(\displaystyle \text{Identity: }\;\sin2\theta \;=\;2\sin\theta\cos\theta\)

\(\displaystyle \text{Substitute }{\bf[1]}:\;\;\sin2\theta \;=\;2\left(\frac{1}{5}\right)\left(\frac{2\sqrt{6}}{5}\right) \;=\;\frac{4\sqrt{6}}{25}\)