Problem 19 - Precalculus with Calculus Previews

Question

In Problems \(19-46,\) find all solutions of the given trigonometric equation if \(x\) is a real number and \(\theta\) is an angle measured in degrees. $$ \cos ^{2} x-1=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = 0 + 360k \) and \( x = 180 + 360k \), where \( k \) is any integer.

Step 1: Understand the equation

The given equation is \( \cos^2 x - 1 = 0 \). Our task is to find all the values of \( x \) that satisfy this equation. This is a standard trigonometric equation.

Step 2: Simplify and Solve for Cosine

Start by rearranging the equation: \( \cos^2 x = 1 \). The square of cosine equals 1 when the cosine function itself is either 1 or -1.

Step 3: Determine the angles for \( \cos x = 1 \)

The cosine of an angle is 1 at \( x = 0 + 360k \) degrees, where \( k \) is any integer. This is because the cosine of \( 0 \) is 1, and cosine has a period of 360 degrees.

Step 4: Determine the angles for \( \cos x = -1 \)

The cosine of an angle is -1 at \( x = 180 + 360k \) degrees, where \( k \) is any integer. This happens because the cosine of \( 180 \) degrees is -1.

Step 5: Combine Solutions

The solutions of the original equation are the solutions where \( \cos x = 1 \) or \( \cos x = -1 \). Therefore, the set of solutions in degrees is \( x = 0 + 360k \) and \( x = 180 + 360k \), where \( k \) is any integer.

Key Concepts

Cosine Function

When studying trigonometric equations, the cosine function is crucial. It is one of the primary trigonometric functions and relates the angle of a right-angled triangle to the ratio of the adjacent side over the hypotenuse. The cosine function is denoted as \( \cos \) and has values that range between -1 and 1 for all real angles. Here are some key points about the cosine function:

• The cosine function is periodic, meaning it repeats its values in regular intervals. Its period is \( 360 \) degrees or \( 2\pi \) radians.
• At \( 0 \) degrees, the value of \( \cos \) is \( 1 \), while at \( 180 \) degrees, the value is \(-1\).
• The cosine function is even, meaning that \( \cos(-x) = \cos(x) \).

Understanding these properties of the cosine function helps us solve equations like \( \cos^2 x - 1 = 0 \).

Angle Measurement

In trigonometry, angles can be measured in various units, but the two most common are degrees and radians. Degrees are a standard unit of angular measurement. A circle is \( 360 \) degrees, which means each degree is \( 1/360^{th} \) of the circle. Here’s how angles play a role in solving trigonometric equations:

• When we say \( x = 0 + 360k \), it means every complete cycle (\( k \) cycles) adds \( 360 \) degrees, resulting in the same cosine value.
• Equations can also involve radians, but in this exercise, emphasis is put on degrees.
• The measurement in degrees helps in intuitively understanding the rotation around a circle.

This unit of measurement is helpful as it divides circles into four quadrants, helping to locate angles quickly in trigonometric functions.

Solution Set

The solution set of a trigonometric equation consists of all the angle values that satisfy the given trigonometric equality. When we solve \( \cos^2 x - 1 = 0 \), this is interpreted by considering the periodic nature of the cosine function.

• For \( \cos x = 1 \), the solutions are \( x = 0 + 360k \) degrees.
• For \( \cos x = -1 \), the solutions are \( x = 180 + 360k \) degrees.

Here, \( k \) represents any integer and accounts for the infinite set of solutions reflecting the periodic behavior of the cosine function. It's important to acknowledge the periodicity as it highlights that the solution can occur multiple times, infinitely, as \( x \) increases or decreases around the circle.

Degrees

Degrees are a unit of angular measurement, which are especially helpful in trigonometry and geometry. In the context of trigonometric equations, degrees make it easier to visualize solutions in terms of rotations or positions along a circle.

• There are \( 360 \) degrees in a full circle, reflecting one complete rotation.
• Particular angles like \( 0 \), \( 90 \), \( 180 \), and \( 270 \) are standard reference angles.
• Using degrees in solutions, like \( x = 0 + 360k \) or \( x = 180 + 360k \), directly ties the periodicity of trigonometric functions to a clear and understandable cycle.

Degrees offer an intuitive way of visualizing where angles occur around the unit circle and help in quickly identifying repetitive patterns in trigonometric equations.