2-3-85 trig identities
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In this problem we're going to use several identities.
Sine of negative Theta is really just negative sine Theta.
If we thought about that in a unit circle, if appears Theta
then down here would be negative Theta and we can see that sine
the height is positive in the unit circle in the first
quadrant and negative in the second.
Or we can look at our Y equals sine X graph and our graph looks
something like this.
And if we put Theta somewhere out here, we can see that that
back here is negative Theta and the Y values are opposite.
So that's a trigger identity we need to become familiar with.
We also know that cosecant repeats every full circle or
every 2π in a unit circle.
So cosecant of Theta is the same thing as cosecant Theta plus 2π.
So to solve this problem, we're going to have negative sine π
fifths times cosecant.
Well, if I have 11 Pi fifths, 2π is really just 10 Pi fifths
because five goes into 10 two times.
So that leaves us a Pi 5th.
Now we also have to remember that cosecant of an angle is
just really one divided by sine of the angle.
It's the reciprocal, it's the multiplicative reciprocal.
So we have negative sine Pi fifth times one over cosecant Pi
5th and when we do the math we can see our final answer there
is going to be -1.