Mathematics: Can someone explain how this answer was produced?

Can someone explain how this answer was produced?

Ответ:

ijohnh14

11.03.2022

Mathematics

Step-by-step 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 $t = \sin(-\frac{11\pi}{4}) = - \sin(\frac{11\pi}{4})$

We know the certain identity.

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

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

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

Another helping identity is the following:

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

$\sin(\frac{3\pi}{4}) = \sin(\pi - \frac{3\pi}{4}) = \sin(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$

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

Now we end up with the following equation.

$\cos^{-1}(-\frac{\sqrt{2}}{2}) = x\\\cos(x) = -\frac{\sqrt{2}}{2} = x = \frac{3\pi}{4}$

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