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Chapter 6: Problem 20

Solve for \(x: \sin ^{-1}\left(\frac{1}{\sqrt{5}}\right)+\cos ^{-1} x=\frac{\pi}{4}\).

Short Answer

Expert verified
The solution to the given equation is \( x = \sqrt{0.7} \)

Step by step solution

01

Write down the equation

Start by writing the equation: \(\sin^{-1} \left( \frac{1}{\sqrt{5}} \right) + \cos^{-1} x = \frac{\pi}{4}\)
02

Isolate the inverse cosine term

Subtract \(\sin^{-1} \left( \frac{1}{\sqrt{5}} \right)\) from both sides of the equation to isolate the term involving \( x \): \(\cos^{-1} x = \frac{\pi}{4} - \sin^{-1} \left( \frac{1}{\sqrt{5}} \right)\)
03

Apply the trigonometric identity

Apply the trigonometric identity \(\sin^{2}(胃) + \cos^{2}(胃) = 1\) and remember that \(\cos^{2}(胃) = 1 - \sin^{2}(胃)\), so that \(\cos 胃 = 卤\sqrt{1 - \sin^{2}(胃)}\). From this, we have \(x = 卤\sqrt{1 - \sin^{2} \left(\frac{\pi}{4} - \sin^{-1} \left( \frac{1}{\sqrt{5}} \right)\right)}\)
04

Simplify the equation

Significant simplification becomes possible if we remember that the square of the inverse sine term \(\sin^{-1} \left( \frac{1}{\sqrt{5}} \right)\) is \(\left( \frac{1}{\sqrt{5}} \right)^{2}\), which equals \( \frac{1}{5} \). Therefore, \(x = 卤\sqrt{1 - \left(\frac{1}{2} - \frac{1}{5}\right)} = 卤\sqrt{\frac{7}{10}} = 卤\sqrt{0.7}\).
05

Determine the range and domain

Recall that the domain (possible values of \( x \)) for the inverse cosine term is such that \( -1 \leq x \leq 1 \). \(\sqrt{0.7}\) is a valid answer falling within the domain, however, \(-\sqrt{0.7}\) makes \(\cos^{-1}x\) undefined. Hence, only one solution exists: \(x = \sqrt{0.7}\).

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Most popular questions from this chapter

The values of \(x\) satisfying \(\tan \left(\sec ^{-1} x\right)=\sin \left(\cos ^{-1}\left(\frac{1}{\sqrt{5}}\right)\right)\) is (a) \(\pm \frac{\sqrt{5}}{3}\) (b) \(\pm \frac{3}{\sqrt{5}}\) (c) \(\pm \frac{\sqrt{3}}{5}\) (d) \(\pm \frac{3}{5}\)

Let \(f(n)=\sum_{k=-n}^{n}\left(\cot ^{-1}\left(\frac{1}{k}\right)-\tan ^{-1}(k)\right)\) such that \(\sum_{n=2}^{10}(f(n)+f(n-1))=a \pi\) then find the value of \((a+1)\).

Let \(m\) be the number of solutions of \(\sin (2 x)+\cos (2 x)+\cos x+1=0\) in \(0

Find the simplest value of \(\cos ^{-1} x+\cos ^{-1}\left(\frac{x}{2}+\frac{\sqrt{3-3 x^{2}}}{2}\right), \forall x \in\left(\frac{1}{2}, 1\right)\)

The number of positive integral solutions of \(\tan ^{-1} x+\cot ^{-1}\left(\frac{1}{y}\right)=\sin ^{-1}\left(\frac{3}{\sqrt{10}}\right)\) is (a) 0 (b) 1 (c) \(\underline{2}\) (d) 3
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