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barxatty [35]
2 years ago
15

Simplify x = cos[2 arcsin(3/5)]. -7/25 7/25 4/5

Mathematics
1 answer:
Marina CMI [18]2 years ago
5 0

Answer:

7/25

Step-by-step explanation:

Let \theta= \arcsin(\frac{3}{5}) so we have x=\cos(2\theta)

As \cos(2\theta)=\cos^2\theta-\sin^2\theta, we'll have \cos[2\arcsin(\frac{3}{5})]=\bigr[\cos(\arcsin(\frac{3}{5}))\bigr]^2-\bigr[(\sin(\arcsin(\frac{3}{5}))\bigr]^2

To determine \cos(\arcsin(\frac{3}{5})), construct a right triangle with an opposite side of 3 and a hypotenuse of 5. This is because since \theta=\arcsin(\frac{3}{5}), then \sin\theta=\frac{3}{5}=\frac{\text{Opposite}}{\text{Hypotenuse}}. If you recognize the Pythagorean Triple 3-4-5, you can figure out that the adjacent side is 4, and thus, \cos\theta=\frac{4}{5}=\frac{\text{Adjacent}}{\text{Hypotenuse}}. This means that \cos(\arcsin(\frac{3}{5}))=\frac{4}{5}.

Hence, \cos[2\arcsin(\frac{3}{5})]=(\frac{4}{5})^2-(\frac{3}{5})^2=\frac{16}{25}-\frac{9}{25}=\frac{7}{25}

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