Finding value of sin2theta

alexedward

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Oct 5, 2010
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If csctheta = 5, 0 < theta < pi/2, find the value of sin2theta. I'm not sure at all where to start with this problem.
 


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:

 
Hello, alexedward!

cscθ=5,    0<θ<π2\displaystyle \csc\theta = 5,\;\;0 < \theta < \tfrac{\pi}{2}

Find the value of sin2θ\displaystyle \sin2\theta

We have:   cscθ  =  51  =  hypopp\displaystyle \text{We have: }\;\csc\theta \;=\;\frac{5}{1} \;=\;\frac{hyp}{opp}

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

Pythagorus says: adj=24=26\displaystyle \text{Pythagorus says: }\:adj = \sqrt{24} = 2\sqrt{6}

Hence: sinθ=15,    cosθ=265    [1]\displaystyle \text{Hence: }\:\sin\theta \,=\,\frac{1}{5},\;\;\cos\theta \,=\,\frac{2\sqrt{6}}{5}\;\;{\bf [1]}


Identity:   sin2θ  =  2sinθcosθ\displaystyle \text{Identity: }\;\sin2\theta \;=\;2\sin\theta\cos\theta

Substitute [1]:    sin2θ  =  2(15)(265)  =  4625\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}

 
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