Problem 114 - 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. $$7 \cos x=4-2 \sin ^{2} x$$

Step-by-Step Solution

Answer

The solutions for the equation within the interval [0,2\(\pi\)] are \( x \approx 1.1593, 5.1239\).

Step 1: Express in terms of a single trigonometric function

Use the identity \( \sin ^{2} x = 1- \cos ^{2} x \) to express the equation in terms of \(\cos x\)\ alone. Plug this identity into the given equation and so \(7 \cos x = 4 - 2 (1-\cos ^{2} x)\) simplifies to \(7 \cos x = 4 - 2 + 2 \cos ^{2} x\), which further simplifies to the quadratic equation \(2 \cos ^{2} x - 7 \cos x + 2 = 0\).

Step 2: Solve the quadratic equation

Solve for \( \cos x \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plugging in the values from our quadratic equation where \(a = 2, b = -7, \text{and } c = 2\), we get that \( \cos x = \frac{7 \pm \sqrt{49 - 16}}{4}\). After simplification, this yields \( \cos x = \frac{7 \pm \sqrt{33}}{4}\). This gives two possible values for \(\cos x\): \( \cos x = \frac{7 + \sqrt{33}}{4} \approx 2.37228\) and \( \cos x = \frac{7 - \sqrt{33}}{4} \approx 0.37772\). However, for the cosine function, the value lies in the range \([-1,1]\), so only \( \cos x = \frac{7 - \sqrt{33}}{4} \) is valid.

Step 3: Solve for x

To find the possible values for \( x \), use the arccosine function, since this is the value such that \( \cos(\text{value}) = \cos x \). Calculate \( x = \arccos(\frac{7 - \sqrt{33}}{4}) \) to obtain \( x \approx 1.1593\) in radians. However, since the cosine function is positive in both the first and fourth quadrants, we also need to consider \( x = 2\pi - \arccos(\frac{7 - \sqrt{33}}{4}) \approx 5.1239\). So, the solutions to the problem within the interval \([0,2\pi)\) are \( x \approx 1.1593, 5.1239\).

Key Concepts

Cosine Function

The cosine function, often written as \( \cos x \), measures the horizontal coordinate of a point on the unit circle corresponding to an angle \( x \). It's one of the fundamental trigonometric functions, cycling between -1 and 1 as \( x \) moves around the circle.

### Characteristics of the Cosine Function

• **Range**: The cosine of any angle will always lie between -1 and 1, i.e., \(-1 \leq \cos x \leq 1\).
• **Periodicity**: It has a period of \(2\pi\), meaning every \(2\pi \) radians, the values repeat.
• **Symmetry**: Cosine is an even function; \( \cos(-x) = \cos(x) \).

These characteristics make the cosine function ideal for modeling periodic phenomena, like sound waves or seasonal patterns. When solving trigonometric equations, knowing its range is crucial for determining valid solutions.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). Solving quadratics is a fundamental skill in algebra that often involves using the quadratic formula.

### Quadratic Formula
The formula to solve for \( x \) in a quadratic equation is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

• **Discriminant**: The expression \( b^2 - 4ac \) under the square root is called the discriminant.
  • If positive, there are two real solutions.
  • If zero, there's exactly one real solution.
  • If negative, solutions are complex numbers.
• **Application in Trigonometry**: When converting trigonometric equations into a quadratic form, like the example with \( 2 \cos^2 x - 7 \cos x + 2 = 0 \), the quadratic formula assists in solving for cosine values directly.

Understanding how to manipulate these equations and solve them precisely plays a key role in solving more complex mathematical problems.

Arccosine Function

The arccosine function, denoted as \( \arccos x \), is the inverse of the cosine function. It finds the angle whose cosine value is \( x \).

### Key Points about the Arccosine Function

• **Domain**: \( \arccos x \) is defined for \(-1 \leq x \leq 1\).
• **Range**: Outputs values in the range \([0, \pi]\), covering angles from 0 to 180 degrees.
• **Use in Trigonometric Solutions**: Handy for finding angles when you know the cosine value, as shown in our trigonometric equation where \( x = \arccos\left( \frac{7 - \sqrt{33}}{4} \right) \).

The arccosine function is especially important when determining all possible solutions. As seen, only considering \( \arccos \) gives one solution, while the full range of cosine reveals another possible angle: \( 2\pi - \arccos x \). Understanding arccosine is essential for finding complete solutions to trigonometric problems.