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Can someone please explain how this answer was produced?

Can someone please explain how this answer was produced?-example-1

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Answer:

Explanation:

First, we know that the sin function is odd which means:

sin(-x) = -sin(x).

Secondly evaluating an inverse trigonometric function with a normal trigonometric function as the argument can be rewritten as an algebraic expression.

Let
t = \sin(-(11\pi)/(4)) = - \sin((11\pi)/(4))

We know the certain identity.


\sin(\theta) = \sin(2\pi + \theta)

We use it to evaluate sin(11 pi / 4).


\sin((11 \pi)/(4)) = \sin({(8\pi)/(4) + (3 \pi)/(4)}) = \sin(2\pi + (3 \pi)/(4)) = \sin((3\pi)/(4))

Another helping identity is the following:


\sin(\theta) = \sin(\pi - \theta)


\sin((3\pi)/(4)) = \sin(\pi - (3\pi)/(4)) = \sin((\pi)/(4)) = (√(2))/(2)

But let's not forget that t = -sin(11 pi/4) = - sqrt(2) / 2

Now we end up with the following equation.


\cos^(-1)(-(√(2))/(2)) = x\\\cos(x) = -(√(2))/(2) => x = (3\pi)/(4)

User Zack Kanter
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