Problem 160 - Algebra and Trigonometry

Question

Solve each equation on the interval \([0,2 \pi)\) Do not use a calculator. $$ 2 \cos x-1+3 \sec x=0 $$

Step-by-Step Solution

Answer

There are no solutions for x on the interval \([0,2\pi)\) in the real number line that satisfy the given equation.

Step 1: Rewrite the equation in terms of cosine

Let's start by noting that \(\sec x = \frac{1}{\cos x}\). So the equation can be rewritten as \(2 \cos x - 1 + \frac{3}{\cos x} = 0\). Then, we proceed to solve the equation for \(\cos x\) by multiplying every term by \(\cos x\) to clear the fraction. We get: \[ 2\cos^2 x - \cos x + 3 = 0 \]

Step 2: Solve the quadratic equation

To solve for \(\cos x\), we treat the equation like a quadratic equation where \(\cos x\) is the variable, with the equation in standard form \(ax^2+bx+c=0\). In this case, \(a=2, b=-1, c=3\). Using the quadratic formula \(x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\), we find:\[\cos x=\frac{1 \pm \sqrt{1^2-4*2*3}}{2*2}\]This simplifies to:\[\cos x=\frac{1 \pm \sqrt{-23}}{4}\] This results in a complex number because the number under the square root is negative. It means no real solutions exist in the form of x, so there could be possibly no angles x in which the original equation satisfies within the interval \(0\leq x<2\pi\)

Step 3: Verify result

It is important to verify if such scenario could occur within the interval \(0 \leq x < 2\pi\). In this interval, the values of cosine function vary between -1 and 1, inclusive. But the equation requires a value of \(\cos x\) which doesn't exist within the real number line. Hence, there isn't an angle x on the unit circle that would provide such result within the interval \(0 \leq x < 2\pi\), essentially confirming that indeed there are no solutions.

Key Concepts

Cosine Function

The cosine function is a fundamental concept in trigonometry. It relates an angle in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. On the unit circle, this function corresponds to the x-coordinate of a point on the circle.

The cosine function is periodic with a period of \(2\pi\). This means it repeats its values every \(2\pi\). Its range is between -1 and 1. Understanding the behavior of the cosine function helps in solving trigonometric equations.

When solving equations involving cosine, recognizing this range is crucial. It helps in identifying whether a particular result is possible within the specified interval (like \(0, 2\pi\) ).

Interval Notation

Interval notation is a way of representing a range of numbers. It’s clear and concise, often used in solutions involving inequalities or trigonometric equations.

For example, the interval \([0, 2\pi)\) denotes all real numbers from 0 to \(2\pi\), including 0 but not \(2\pi\).

This notation helps specify where the solutions to an equation should lie. Using interval notation simplifies the communication of which part of the number line we are interested in, especially in trigonometric contexts. It’s essential to accurately read and interpret this notation to find and validate solutions.

Quadratic Equations

Quadratic equations are polynomials of the form \(ax^2 + bx + c = 0\). They show up often in algebra and trigonometry. In this problem, the equation was rewritten in a quadratic form using cosine.

Solving a quadratic equation might involve the use of the quadratic formula:

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

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation.
• The square root term, \(b^2 - 4ac\), is the discriminant and determines the nature of the roots.

If the discriminant is negative, as it was in the example with \(-23\), the roots are complex and not real. This indicates there are no real solutions, important for solving equations within specific intervals.

Secant Function

The secant function is the reciprocal of the cosine function, expressed as \(\sec x = \frac{1}{\cos x}\).

This function has a range where \(\cos x\) cannot be zero because you cannot divide by zero. Understanding secant is crucial when transforming trigonometric equations into a form easier to solve, such as quadratics.

The secant function helps extend the range of possibilities for solutions by considering reciprocal relationships. Still, you'd apply it under constraints where cosine is defined, typically avoiding angles where cosine equals zero. This reciprocal understanding aids in rewriting and solving complex equations, leading to insights on possible solution existence or non-existence within specified intervals. Understanding secant's restrictions ensures accurate trigonometric equation solutions.