Problem 112 - Precalculus Essentials

Question

Use the most appropriate method to solve each equation on the interval \([0,2 \pi) .\) Use exact values where possible or give approximate solutions correct to four decimal places. $$\cos ^{2} x+5 \cos x-1=0$$

Step-by-Step Solution

Answer

With each of these steps, a complete solution can be arrived upon by using the quadratic formula first and then applying the inverse cosine function.

Step 1: Recognize as a Quadratic Equation

Reformulate the equation '\( \cos ^2 x + 5 \cos x - 1 = 0 \)' as a quadratic equation by substituting \( u = \cos x \). Thus, the equation becomes \( u^2 + 5u - 1 = 0 \).

Step 2: Apply Quadratic Formula

Apply the quadratic formula \( u = \frac{-b ± \sqrt{b^2 - 4ac}}{2a} \), where a = 1, b = 5, and c = -1. Solve for 'u' which gives you two solutions \( u_1 \) and \( u_2 \).

Step 3: Reconvert 'u' Back to (\( \cos x \))

Reconvert 'u' back to \( \cos x \) by finding \( x = \arccos(u) \). This will give you two different solutions in terms of x. Thus, we have \( x_1 = \arccos(u_1) \) and \( x_2 = \arccos(u_2) \).

Step 4: Verify the Solutions

Verify that the solutions are in the interval [0, \(2\pi\)] and modify them as required. Only then, the solutions will meet the requirements of the exercise.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our context, the equation \( \cos^2 x + 5 \cos x - 1 = 0 \) is a quadratic equation in terms of \( \cos x \). By substituting \( u = \cos x \), the equation becomes \( u^2 + 5u - 1 = 0 \). This transformation allows us to apply methods used for solving quadratic equations on trigonometric problems.

• Recognizing such transformations simplifies solving trigonometric equations.
• Solving for \( u \) first avoids complicated trigonometric manipulations.

This insight helps in tackling seemingly complex trigonometric equations with relative ease.

Interval Notation

Interval notation is a way of representing the set of all numbers between two endpoints. For example, the interval \([0, 2\pi)\) includes all numbers starting from \(0\) up to but not including \(2\pi\).

• Brackets \([\) and \(]\) indicate inclusive bounds, meaning the endpoint is part of the interval.
• Parentheses \((\) and \()\) mean the endpoint is not included.

When solving trigonometric equations, knowing the correct interval to work within ensures solutions fall within the expected range. In trigonometric contexts, \([0, 2\pi)\) usually represents one full rotation on the unit circle, outlining the standard range for angles in radians.

Arccos

The arccosine function, written as \( \arccos(x) \), is the inverse of the cosine function. It answers the question, "What angle has this cosine value?" When you have a value \( u \) and you need to find the angle \( x \), you use \( \arccos(u) \).

• Values of \( u \) must be within the range \([-1, 1]\) for \( \arccos \) to provide a real number result.
• The output of \( \arccos \) is always within the interval \([0, \pi]\).

Hence, after solving the quadratic equation with substitutions, you find \( x \) using \( \arccos \), which properly converts the result back into a trigonometric angle.

Quadratic Formula

The quadratic formula is a universal method for finding the solutions of any quadratic equation \( ax^2 + bx + c = 0 \). It is expressed as:\[u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( u \) represents the variable such as \( u \) in our context. The formula gives two solutions, which can be real or complex.

• The discriminant \( b^2 - 4ac \) determines the nature of the roots (real or complex).
• If the discriminant is positive, there are two distinct real solutions.
• If it is zero, there are two identical real solutions.

The quadratic formula is particularly helpful when factorization is difficult, or the coefficients do not lend themselves to simpler solution methods.