Problem 73 - Precalculus Essentials

Question

Use an identity to solve each equation on the interval \([0,2 \pi)\) $$\cos 2 x+5 \cos x+3=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = 2\pi\).

Step 1: Use of Double Angle Identity

Use the double angle identity \(\cos 2x = 1 - 2 cos^{2}x\). Substituting into the equation gives \(1 - 2\cos^{2}x + 5\cos x + 3 = 0\). Simplifying this gives \(2 \cos^{2}x - 5\cos x + 2 = 0\). Notice that we have now expressed the original equation as a quadratic equation in terms of \(\cos x\).

Step 2: Apply the Quadratic Formula

We can now apply the quadratic formula to solve for \(\cos x\). The quadratic formula is \(-b \pm \sqrt {b^{2}-4ac} \over 2a\), where \(a = 2\), \(b = -5\), and \(c = 2\) in our equation. This gives us \(\cos x = 1, \cos x = 2\).

Step 3: Find the Value(s) of \(x\)

We now find for what values of \(x\) we get \(\cos x = 1\) and \(\cos x = 2\). Solving the first equation \(\cos x = 1\) gives two solutions \(x = 0\) and \(x = 2\pi\), both fall within the interval \([0,2\pi)\). However, \(\cos x = 2\) has no real solutions because the value of cosine function lies between -1 and 1 for real values of \(x\).

Key Concepts

Double Angle Identity

The double angle identity is a fundamental concept in trigonometry that can simplify expressions involving angles. Specifically, the identity for cosine is given by \( \cos 2x = 1 - 2 \cos^2 x \) or \( \cos 2x = 2 \cos^2 x - 1 \). Each variant can be used depending on the problem you are trying to solve. In the original exercise, the identity \( \cos 2x = 1 - 2 \cos^2 x \) was used to transform the equation. By substituting it, the expression involves only \( \cos x \), making it easier to solve. This technique is especially helpful when tackling quadratic equations involving trigonometric functions.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of a quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:

• \(-b \pm \sqrt{b^2 - 4ac} \over 2a\)

To use the quadratic formula, identify the coefficients \( a, b, \) and \( c \). In our exercise, after using the double angle identity and simplifying, we got \( 2 \cos^2 x - 5 \cos x + 2 = 0 \). The values for coefficients are: \( a = 2 \), \( b = -5 \), \( c = 2 \). By applying these values into the formula, we solve for \( \cos x \). In this particular case, we found solutions: \( \cos x = 1 \) and \( \cos x = 2 \). Be cautious, as we need to make sure solutions are within the valid range of the cosine function, which is between \(-1\) and \(1\).

Interval Notation

Interval notation is a way to specify a range of values and is often used in mathematics to indicate the domain or solution set for equations. Careful consideration of interval notation ensures the solutions fall within a given range. In our problem, the interval \([0, 2\pi)\) was specified. This interval includes all numbers from \(0\) to \(2\pi\) but not including \(2\pi\). In the example given, \( x = 0 \) was found as a valid solution for \( \cos x = 1 \), but \( x = 2\pi \) was not included because the interval does not include \(2\pi\). Understanding how to properly apply interval notation is crucial when ensuring solutions are valid for the given domain.