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Chapter 1: Problem 61

Draw a right triangle to simplify the given expressions. Assume \(x>0\) $$\cos \left(\sin ^{-1} x\right)$$

Short Answer

Expert verified
The simplified expression for \(\cos(\sin^{-1}x)\) is \(\frac{\sqrt{c^2 - a^2x^2}}{c}\).

Step by step solution

01

Create a right triangle

Since sin function is the ratio of the opposite side to the hypotenuse, let's create a right triangle with legs \(a\) and \(b\), and hypotenuse \(c\). Now let's assume that the opposite side has length \(ax\) and the hypotenuse has length \(c\) where \(x>0\), so our sine ratio becomes: \(\sin\theta = \frac{ax}{c}\).
02

Find the length of the adjacent side

By the Pythagorean theorem, we have \(a^2 + b^2 = c^2\). We're given that \(\sin\theta = \frac{ax}{c}\), so the opposite side is \(ax\). To find the adjacent side, we can use the Pythagorean theorem and solve for \(b\): $$b^2 = c^2 - a^2x^2.$$
03

Find the cosine of the angle

The cosine function gives us the ratio of the adjacent side to the hypotenuse. Using the lengths that we have found in the previous steps, we can calculate the cosine ratio as \(\cos\theta = \frac{b}{c}\). Since we know \(b^2 = c^2 - a^2x^2\) and \(c>0\), we can substitute this into the equation: $$\cos\theta = \frac{\sqrt{c^2 - a^2x^2}}{c}.$$
04

Simplify the expression

The original expression was \(\cos(\sin^{-1}x)\). Now, we know that \(\cos\theta = \frac{\sqrt{c^2 - a^2x^2}}{c}\). To find the final simplified expression, we need to find the cosine of the angle whose sine ratio is \(x\). So we can substitute this result back into the given expression: $$\cos(\sin^{-1}x) = \frac{\sqrt{c^2 - a^2x^2}}{c}.$$ Since we don't have any further information about the specific values of \(a\) and \(c\), this is the final simplified expression for the problem.

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Most popular questions from this chapter

A light block hangs at rest from the end of a spring when it is pulled down \(10 \mathrm{cm}\) and released. Assume the block oscillates with an amplitude of \(10 \mathrm{cm}\) on either side of its rest position and with a period of 1.5 s. Find a function \(d(t)\) that gives the displacement of the block \(t\) seconds after it is released, where \(d(t)>0\) represents downward displacement.

Without using a calculator, evaluate or simplify the following expressions. $$\sec ^{-1} 2$$

a. Let \(g(x)=2 x+3\) and \(h(x)=x^{3} .\) Consider the composite function \(f(x)=g(h(x))\). Find \(f^{-1}\) directly and then express it in terms of \(g^{-1}\) and \(h^{-1}\). b. Let \(g(x)=x^{2}+1\) and \(h(x)=\sqrt{x} .\) Consider the composite function \(f(x)=g(h(x))\). Find \(f^{-1}\) directly and then express it in terms of \(g^{-1}\) and \(h^{-1}\). c. Explain why if \(g\) and \(h\) are one-to-one, the inverse of \(f(x)=g(h(x))\) exists.

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