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Chapter 5: Problem 72

Use inverse functions where needed to find all solutions of the equation in the interval \(\mathbf{0}, \mathbf{2} \boldsymbol{\pi}\) ). $$\sec ^{2} x+2 \sec x-8=0$$

Short Answer

Expert verified
The solutions to the trigonometric equation \(\sec ^{2} x+2 \sec x-8=0\) in the interval from 0 to \(2\pi\) are \(x \approx 1.318, 2.094, 4.188, 5.823\).

Step by step solution

01

Convert sec to cos

Express the equation in terms of the cosine function. Substitute \(\sec(x)\) with \(1/\cos(x)\) in the original equation to yield: \((1/\cos^2(x)) + 2(1/\cos(x)) - 8 = 0\)
02

Clear the Fraction

Get rid of the denominators by multiplying through by \(\cos^2(x)\), resulting in the new equation: \(1 + 2\cos(x) - 8\cos^2(x) = 0\).
03

Rearrange the Equation

Rearrange the equation to the standard quadratic form \(ax^2 + bx + c = 0\), which gives: \(8\cos^2(x) - 2\cos(x) - 1 = 0\).
04

Solve for cos(x)

Solve the equation for \(\cos(x)\) using the quadratic formula \(x = [ -b \pm \sqrt{ (b^2 - 4ac) } ] / 2a\), to obtain the roots: \(\cos(x) = 1/4, -1/2\). Determine the corresponding x-values within the interval from 0 to \(2\pi\).
05

Solve for x

Invoke the inverse cosine function, \(\cos^{-1}\), to solve for x. The solutions to the original equation \(x = \cos^{-1}(1/4)\) and \(x = \cos^{-1}(-1/2)\) are \(x \approx 1.318, 2.094, 4.188, 5.823\) in the range of 0 to \(2\pi\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Inverse Functions
Inverse functions play a key role in solving equations where the solution involves finding the original angle from a known trigonometric ratio. In this exercise, after identifying the values of \( \cos(x) \) using the quadratic equation, the inverse cosine function \( \cos^{-1} \) is applied. This function helps unravel the angle corresponding to a specific cosine value. The inverse squared secant, \( \sec^{-2} \), would allow us to derive \( \cos(x) \), but it's more common to switch directly to \( \cos(x) \) and use \( \cos^{-1} \) instead.

  • Key Point: \( \cos^{-1} \) helps in translating between cosine values and their associated angle measures.
  • Ensures all potential angle solutions within the specified interval are found.
  • Helps confirm the angles that satisfy \( \cos(x) = \frac{1}{4} \) and \( \cos(x) = -\frac{1}{2} \) in the interval \([0, 2\pi]\).
Quadratic Equation
Quadratic equations are essential when trigonometric equations are expressed in polynomial forms involving squares and linear terms. Here, the equation is transformed into a standard quadratic form: \( 8\cos^2(x) - 2\cos(x) - 1 = 0 \). Solving this involves using the well-known quadratic formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

  • Identify coefficients: Here, \( a = 8 \), \( b = -2 \), and \( c = -1 \).
  • Calculate the root values of \( \cos(x) \).
  • Find roots: \( \cos(x) \) values like \( \frac{1}{4} \) and \( -\frac{1}{2} \).
Transforming trigonometric functions into a quadratic form often makes them easier to solve and interpret.
Cosine Function
The cosine function, \( \cos(x) \), is fundamentally important in trigonometry. It measures the adjacent side over the hypotenuse in a right-angled triangle or it gives the x-coordinate of a point on the unit circle, depending on context. In this context, we first convert the secant into cosine, understanding that \( \sec(x) = \frac{1}{\cos(x)} \).

  • Transformation: Using the relationship between secant and cosine helps simplify the equation.
  • Range Consideration: Ensure \( \cos(x) \) falls within its standard range of \([-1, 1]\).
  • Determine feasible \( x \) values that solve \( \cos(x) = \frac{1}{4} \) and \( -\frac{1}{2} \)
Converting secant to cosine reduces complexity and clarifies the trigonometric relationships involved.
Secant Function
The secant function, denoted \( \sec(x) \), is the reciprocal of the cosine function: \( \sec(x) = \frac{1}{\cos(x)} \). It is less commonly used directly than the sine and cosine functions, but serves important roles in solving trigonometric equations by relating to cosine. For this problem:

  • Initial Conversion: Transform \( \sec(x) \) to \( \frac{1}{\cos(x)} \) to bring it in terms of the more manageable cosine form.
  • Compatibility: Ensure the function's range aligns with real values of cosine.
  • Emphasizes the reciprocal relationship between \( \sec \) and \( \cos \).
This transformation allows easier manipulation of equations and leads to finding feasible solutions for the initial equation.

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Most popular questions from this chapter

The range of a projectile fired at an angle \(\theta\) with the horizontal and with an initial velocity of \(v_{0}\) feet per second is \(r=\frac{1}{32} v_{0}^{2} \sin 2 \theta\) where \(r\) is measured in feet. An athlete throws a javelin at 75 feet per second. At what angle must the athlete throw the javelin so that the javelin travels 130 feet?

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Explain in your own words how knowledge of algebra is important when solving trigonometric equations.

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