Problem 63 - Algebra 2

Question

Find the \(x\) -intercepts of the graph of each function. $$ y=2 \cos ^{2} \theta-3 \cos \theta-2 $$

Step-by-Step Solution

Answer

The x-intercepts of the function \(y=2 \cos ^{2} \theta-3 \cos \theta-2\) are \(x = 2\pi/3, 4\pi/3\).

Step 1: Recognize Quadratic Equation

Recognize the given equation \(0=2 cos^2(x) - 3cos (x) - 2\) as a quadratic equation in terms of \(cos(x)\), namely \(0 = 2u^2 - 3u - 2\) where \(u = cos(x)\).

Step 2: Solve Quadratic Equation Using Quadratic Formula

Solve for \(u\) using the quadratic formula \(u = (-b \pm \sqrt{b^2 - 4ac}) / 2a\). This gives \(u = (3 \pm \sqrt{9+16}) / 4 = (3 \pm 5) / 4\). This results in two solutions: \(u = -0.5, 2\).

Step 3: Solve for the values of \(x\)

Equating each value of \(u\) to \(cos(x)\), we can solve for the corresponding values of \(x\). For \(u = -0.5\), \(cos(x) = -0.5\) which gives \(x = 2\pi/3, 4\pi/3\) (in the basic interval \([0,2\pi]\)). For \(u = 2\), there are no real solutions since \(-1 \leq cos(x) \leq 1\) for all \(x\). Thus, the x-intercepts of the function are \(2\pi/3, 4\pi/3\).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation of the form \(ax^2 + bx + c = 0\). They describe a parabolic graph and can have up to two solutions called roots. In the exercise, we worked with a quadratic form, but instead of \(x\), we had \(\cos(\theta)\). This is common in trigonometry, where we transform complicated trigonometric expressions into polynomial-like forms for easier solving. By substituting \(u = \cos(\theta)\), the original equation turned into a more familiar quadratic equation \(0 = 2u^2 - 3u - 2\), allowing us to utilize methods like the quadratic formula to find the solutions.

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Cosine is periodic, repeating every \(2\pi\) radians, and has a range of values from -1 to 1. In our exercise, solving for \(\cos(x)\) helped us identify the \(x\)-values where these values matched our solutions from the quadratic equation. Recognizing the characteristics of the cosine graph such as its amplitude, period, and phase shift, is crucial when determining where on the unit circle specific values like \(-0.5\) occur.

x-intercepts

Finding \(x\)-intercepts, or roots, in a trigonometric function graph is about determining the points at which the graph crosses the x-axis (\(y = 0\)). For the equation \(y = 2\cos^2(\theta) - 3\cos(\theta) - 2\), finding these points involved solving the equation in terms of \(\cos(\theta)\) and then translating those into \(x\)-coordinate intersections. The \(x\)-intercepts \(2\pi/3\) and \(4\pi/3\) emerged because these are the angles where \(\cos(x) = -0.5\), while for \(u = 2\), there were no solutions since \(\cos(x) \) must remain between -1 and 1.

Trigonometric Identities

Trigonometric identities are equations that are always true for every value of the variable within the domain of the trigonometric function. These identities are essential tools in solving and simplifying expressions. Common identities include the Pythagorean identity and angle sum formulas. In our problem, these identities were implicitly utilized to solve for \(\cos(x)\) when transforming the function into its quadratic form. Understanding and applying these identities allow for simplification and solving of complex equations, simplifying them into forms that are easier to solve, just like transforming to \(0 = 2u^2 - 3u - 2\). Recognizing and applying these identities is a fundamental skill in both algebra and trigonometry.