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Chapter 5: Problem 38

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

Short Answer

Expert verified
The identity expressing \(\tan\left(\sin^{-1} t\right)\) as a function of \(t\) is: \[\tan\left(\sin^{-1} t\right) = \frac{t}{\sqrt{1 - t^2}}\]

Step by step solution

01

Express the angle in terms of a right triangle

Let us consider a right triangle ABC, where angle B = 90 degrees, angle A = \(\sin^{-1} t\), and angle C is the remaining angle. According to the definition of sine, we have \(\sin A = \frac{BC}{AC} = t\). Let's say BC = t, and AC = 1.
02

Find the length of side AB

Using Pythagorean theorem, \[\begin{aligned} AB^2 &= AC^2 - BC^2 \\\\ AB^2 &= 1^2 - t^2 \\\\ AB &= \sqrt{1 - t^2} \end{aligned}\]
03

Calculate the tangent of angle A

Now, we'll find the tangent of angle A by using the definition of tangent in a right triangle. \[\begin{aligned} \tan A &= \frac{BC}{AB} \\\\ &= \frac{t}{\sqrt{1 - t^2}} \end{aligned}\]
04

Substitute the expression for angle in the given function

Now that we have found the tangent for angle A (which equals \(\sin^{-1} t\)), we can easily substitute it in the given expression: \[\tan\left(\sin^{-1} t\right) = \frac{t}{\sqrt{1 - t^2}}\] Therefore, the identity expressing \(\tan\left(\sin^{-1} t\right)\) as a function of \(t\) is: \[\tan\left(\sin^{-1} t\right) = \frac{t}{\sqrt{1 - t^2}}\]

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Most popular questions from this chapter

Show that $$\tan ^{2}(2 x)=\frac{4\left(\cos ^{2} x-\cos ^{4} x\right)}{\left(2 \cos ^{2} x-1\right)^{2}}$$ for all numbers \(x\) except odd multiples of \(\frac{\pi}{4}\).

Find exact expressions for the indicated quantities. The following information will be useful: $$ \begin{array}{l} \cos 22.5^{\circ}=\frac{\sqrt{2+\sqrt{2}}}{2} \text { and } \sin 22.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2} \\ \cos 18^{\circ}=\sqrt{\frac{\sqrt{5}+5}{8}} \text { and } \sin 18^{\circ}=\frac{\sqrt{5}-1}{4} \end{array} $$ [The value for \(\sin 22.5^{\circ}\) used here was derived in Example 4 in Section \(5.5 ;\) the other values were derived in Exercise 64 and Problems 102 and 103 in Section \(5.5 .]\) $$\sin 37.5^{\circ}$$

Show that if \(\cos (2 u)=\cos (2 v),\) then \(|\cos u|=\) \(|\cos v|\).

Show that $$\cos \frac{\pi}{32}=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}$$ [Hint: First do Exercise 66.]

Evaluate the indicated expressions assuming that $$ \begin{array}{ll} \cos x=\frac{1}{3} & \text { and } \sin y=\frac{1}{4} \\ \sin u=\frac{2}{3} & \text { and } \cos v=\frac{1}{5} \end{array} $$ Assume also that \(x\) and \(u\) are in the interval \(\left(0, \frac{\pi}{2}\right),\) that \(y\) is in the interval \(\left(\frac{\pi}{2}, \pi\right),\) and that \(v\) is in the interval \(\left(-\frac{\pi}{2}, 0\right) .\) $$\sin (x+y)$$
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