Problem 38 - Precalculus

Question

For the following exercises, let \(f(x)=\cos x\) On \([0,2 \pi),\) solve the equation \(f(x)=\cos x=0\)

Step-by-Step Solution

Answer

Solutions are \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

Step 1: Understand the Function

The function given is \( f(x) = \cos x \). This is a trigonometric function representing the cosine of the angle \( x \), where \( x \) is measured in radians.

Step 2: Identify the Interval

We are asked to solve the equation on the interval \( [0, 2\pi) \). This means we are considering values of \( x \) starting from 0 up to but not including \( 2\pi \).

Step 3: Set the Equation to Zero

The equation to solve is \( \cos x = 0 \). Our task is to find where the cosine of the angle is zero within the specified interval.

Step 4: Solve for x

\( \cos x \) equals zero at angles where \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) within the interval \( [0, 2\pi) \). These are the angles commonly known as 90 degrees and 270 degrees, converted to radians.

Step 5: Verify the Solutions

Check that \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) satisfy \( \cos x = 0 \) and belong to the interval \( [0, 2\pi) \). Both values do make \( \cos x = 0 \) and lie within the interval.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions, which is crucial in understanding wave patterns, oscillations, and circular movements. It's particularly interesting due to the symmetrical properties it exhibits on the unit circle.

• Understanding \( \cos x \): This function measures the horizontal component of a point on the unit circle corresponding to an angle \( x \). The values of \( \cos x \) range between -1 and 1.
• Graphical Representation: The graph of the cosine function starts at \( y=1 \) when \( x=0 \), and it oscillates between its maximum at 1 and its minimum at -1, crossing 0 at certain intervals.
• Significance of Zero crossings: In problems like \( \cos x = 0 \), these are the points where the cosine graph intersects the x-axis. These crossings help identify specific angle values important for periodic functions in various applications.

Keep in mind that solving \( \cos x = 0 \) focuses on finding where the graph crosses the x-axis within a given interval.

Radian Measure

Radians offer a natural way to measure angles based on the radius of a circle. Instead of using degrees, which divide a circle into 360 parts, radians use the circle's own radius as the unit length.

• Definition: One radian is the angle at the center of a circle that subtends an arc equal in length to the radius of the circle.
• Conversion to Degrees: To convert from radians to degrees, use the equation \( 1 \, \text{radian} = \frac{180}{\pi} \, \text{degrees} \). So, for example, \( \pi \) radians equals 180 degrees.
• Using Radians in the Unit Circle: This measure makes calculations involving the circle's circumference straightforward, given that an entire revolution (\( 360 \text{ degrees} \)) is equivalent to \( 2\pi \). This is why trigonometric functions often prefer radian measure over degrees.

In trigonometric equations, interpreting angles in radians simplifies mathematical expressions and is crucial for calculus applications.

Unit Circle

The unit circle is a fundamental concept in trigonometry, used to define the sine, cosine, and tangent functions. It is a circle with a radius of one centered at the origin of the coordinate plane.

• Standard Position: On the unit circle, any angle \( x \) is typically measured from the positive x-axis, counterclockwise. This standard position helps in visualizing solutions to trigonometric equations.
• Coordinates as Function Values: Points on the unit circle are expressed as \( (\cos x, \sin x) \), meaning the x-coordinate corresponds to the cosine value, and the y-coordinate to the sine value of the angle.
• Finding Zeroes: For \( \cos x = 0 \), the solutions are angles where the unit circle intersects the y-axis, such as \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \), which correspond to points \( (0,1) \) and \( (0,-1) \) respectively.

Effective use of the unit circle allows for a geometric understanding of trigonometric functions, providing valuable insight into the behavior of these functions over different intervals.