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Chapter 5: Problem 71

Use inverse functions where needed to find all solutions of the equation in the interval \([0,2 \pi)\). $$\sec ^{2} x-4 \sec x=0$$

Short Answer

Expert verified
The solutions to the equation within the interval [0,2蟺) are \(x = \arccos(1/4)\) and \(x = 2\pi - \arccos(1/4)\).

Step by step solution

01

Rewrite the equation to a quadratic form

Try to write the equation in terms of u by substituting \(u = \sec x\). So we get \(u^{2} - 4u = 0\)
02

Factorise the expression

Next, factorise the expression to solve for u. Thus, the quadratic equation \(u^{2} - 4u = 0\) becomes \(u(u - 4) = 0\)
03

Solve for u

Set each factor equal to zero and solve for u: \(u = 0\) or \(u - 4 = 0\). This gives two roots, u = 0 and u = 4.
04

Substitute u back with sec(x)

Since \(u = \sec(x)\), substitute u with \(\sec(x)\) to get the equations: \(\sec(x) = 0\) and \(\sec(x) = 4\).
05

Apply the definition of sec(x)

Applying the definition \(\sec(x) = 1/ \cos(x)\), we get the two equations \(1/ \cos(x) = 0\) and \(1/ \cos(x) = 4\). Therefore, we solve the following equations for x: \(\cos(x) = \infty\) and \(\cos(x) = 1/4\).
06

Solving the equations

\(\cos(x) = \infty\) has no solution since the cosine function ranges between -1 and 1. For the equation \(\cos(x) = 1/4\), obtain the values of x within the range [0, 2蟺) using the arccos function or by graph analysis.
07

Final solution

Solving \(\cos(x) = 1/4\) on the interval [0, 2蟺), we obtain two solutions: \(x = \arccos(1/4)\) and \(x = 2\pi - \arccos(1/4)\).

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