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nevsk [136]
3 years ago
12

Find the exact value of the composite function show work​

Mathematics
1 answer:
belka [17]3 years ago
3 0

Answer:

\frac{2x}{\sqrt{4x^{2} -1} }

Step-by-step explanation:

This function can be written as:

\frac{1}{cos(sin^{-1}(\frac{1}{2x}))  }

sin^{-1}(\frac{1}{2x}) means the angle whose sine is 1/(2x).  This implies that the denominator is the cosine of the angle whose sine is 1/(2x)

Think of a right triangle and consider the angle with this sine.  Then the opposite side has length 1 and the hypotenuse has length  2x.  The Pythagorean theorem then implies that the adjacent side is

\sqrt{4x^{2} -1}

The cosine of the angle is adjacent over hypotenuse, so the cosine is

\frac{\sqrt{4x^{2} -1}}{2x}

The secant is the inverse of the cosine, so the answer is

\frac{2x}{\sqrt{4x^{2} -1} }

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