/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 28 Find an identity expressing \(\s... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 28

Find an identity expressing \(\sin \left(\cos ^{-1} t\right)\) as a nice function of \(t\).

Short Answer

Expert verified
The identity expressing \(\sin(\cos ^{-1}(t))\) as a function of \(t\) is: \(\sin(\cos ^{-1}(t)) = \sqrt{1 - t^2}\)

Step by step solution

01

Define a new variable

Let's define a new variable, let \(x\) be equal to the inverse cosine function of \(t\): \(x = \cos^{-1}(t)\)
02

Express x in terms of cosine

According to our definition, \(x = \cos^{-1}(t)\), we can write the cosine of \(x\) as: \(\cos(x) = t\)
03

Use the Pythagorean identity to find sine of x

We know that the Pythagorean identity states that: \(\sin^2(x) + \cos^2(x) = 1\) Since we know the value of \(\cos(x)\), we can find the value of \(\sin(x)\) by substituting \(\cos(x) = t\) into the Pythagorean identity: \(\sin^2(x) + t^2 = 1\)
04

Solve for sin(x)

Solving for \(\sin(x)\) from the equation above: \(\sin(x) = \pm \sqrt{1 - t^2}\) However, since the range of the inverse cosine function is \(0 \leq x \leq \pi\) and sine is non-negative in this range, we can safely say that the sine of x will be positive: \(\sin(x) = \sqrt{1 - t^2}\)
05

Substitute the value of x

Recall that \(x = \cos^{-1}(t)\). This means that we can substitute this value back into our equation for sine of x: \(\sin(\cos ^{-1}(t)) = \sqrt{1 - t^2}\) Hence, the identity expressing \(\sin(\cos^{-1}(t))\) as a function of \(t\) is: \(\sin(\cos ^{-1}(t)) = \sqrt{1 - t^2}\)

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