Problem 26 - Algebra 2 and Trigonometry

Question

If \(\mathrm{f}(x)=\cos 2 x,\) find \(\mathrm{f}\left(\frac{3 \pi}{4}\right)\)

Step-by-Step Solution

Answer

The value of \(f\left(\frac{3\pi}{4}\right)\) is 0.

Step 1: Understand the Function

We are given the function \(f(x) = \cos 2x\). This function takes an input \(x\), doubles it, and then takes the cosine of the result.

Step 2: Substitute the Value

We need to find \(f\left(\frac{3\pi}{4}\right)\). Substitute \(\frac{3\pi}{4}\) for \(x\) in the expression for \(f(x)\). This gives \(f\left(\frac{3\pi}{4}\right) = \cos\left(2 \times \frac{3\pi}{4}\right)\).

Step 3: Simplify the Argument of Cosine

Calculate \(2 \times \frac{3\pi}{4}\), which simplifies to \(\frac{6\pi}{4}\) or \(\frac{3\pi}{2}\). Thus, we have \(f\left(\frac{3\pi}{4}\right) = \cos\left(\frac{3\pi}{2}\right)\).

Step 4: Evaluate the Cosine Function

Recall from trigonometric knowledge that \(\cos\left(\frac{3\pi}{2}\right) = 0\) because \(\frac{3\pi}{2}\) corresponds to 270 degrees on the unit circle, where the cosine value is 0.

Key Concepts

Cosine Function

The cosine function, denoted as \(\cos(x)\), is a fundamental aspect of trigonometry. Originating from the unit circle, where the circle has a radius of 1, the cosine function helps us understand the horizontal distance from the origin to a point on the circle.
The cosine function has certain basic properties:

• It is periodic with a period of \(2\pi\), meaning that the function repeats its values every \(2\pi\) units.
• The range of the cosine function is from -1 to 1.
• It is an even function, meaning \(\cos(-x) = \cos(x)\).

The function \(f(x) = \cos(2x)\) involves a transformation of the basic cosine function where the angle \(x\) is doubled.
Thus, the period becomes \(\pi\) (half of the original period). This doubling transformation compresses the standard cosine wave, making it oscillate faster.

Unit Circle

The unit circle is a circle with a radius of one, centered at the origin of a coordinate plane. It's a crucial tool in trigonometry because it provides a geometric representation of the trigonometric functions, including sine and cosine.
Here's how it works:

• Every point on the unit circle represents an angle \(\theta\) with the positive x-axis.
• The x-coordinate of a point on the unit circle is \(\cos(\theta)\), while the y-coordinate is \(\sin(\theta)\).
• The angle is typically measured in radians, moving counterclockwise from the positive x-axis.

In our exercise, \(\frac{3\pi}{2}\) radians corresponds to 270 degrees on the unit circle. At this angle, the point is (0, -1), thus resulting in a cosine value of 0. This symmetry and positioning on the unit circle help comprehend why certain angles have specific cosine values.

Angle Conversion

In trigonometry, understanding angles often hinges on converting between degrees and radians. This is essential when working with trigonometric functions because these functions usually operate on radian measures.
To convert degrees to radians:

• Use the relation \(\pi\) radians = 180 degrees.
• Multiply the degree measure by \(\frac{\pi}{180}\) to get the radian measure.

For example, 270 degrees can be converted by:\[270 \times \frac{\pi}{180} = \frac{3\pi}{2}\]Conversely, converting from radians to degrees involves:

• Multiply the radian measure by \(\frac{180}{\pi}\).

This knowledge is key in the given exercise when interpreting the angle \(\frac{3\pi}{2}\) radians as 270 degrees on the unit circle, thereby confirming the cosine function yields a value of 0.