Problem 39 - Precalculus: A Concise Course

Question

Evaluate the trigonometric function using its period as an aid. $$ \cos \frac{17 \pi}{4} $$

Step-by-Step Solution

Answer

\(\cos \frac{17 \pi}{4} = \sqrt{2}/2\)

Step 1: Identify the period of cosine

Trigonometric functions such as cosine have a period of \(2\pi\). This means that for every \(2\pi\) radians (or 360 degrees), the function values repeat. Therefore, to simplify \(\cos \frac{17 \pi}{4}\), we will first find an equivalent angle within one full revolution which is inside the interval between 0 and \(2\pi\).

Step 2: Simplify the argument

To simplify the argument \(\frac{17 \pi}{4}\) to within one cycle of the cosine function, we can subtract multiples of \(2\pi\) from the argument until it comes within the interval [0, \(2\pi\)]. In this case, since each full circle is \(2\pi\) or \(8\pi/4\), we subtract \(2 \times 8\pi/4\) from \(17\pi/4\) to get \(17\pi/4 - 2 \times 8\pi/4 = \pi/4\).

Step 3: Evaluate the cosine function

Finally, we compute the value of the cosine function using the simplified argument. The cosine of \(\pi/4\) is a known value from the unit circle, which is \(\sqrt{2}/2\). Therefore, \(\cos \frac{17 \pi}{4} = \cos(\pi/4) = \sqrt{2}/2\).

Key Concepts

Period of Trigonometric Functions

Trigonometric functions are fascinating because they repeat their values at regular intervals. This characteristic is known as the 'period' of the function.
For instance, the period of the cosine function is exactly \(2\pi\) radians (or 360 degrees).
What this essentially means is that every \(2\pi\) units along the x-axis, the cosine function starts repeating its pattern of outputs.
This periodic nature is essential not only in simplifying trigonometric computations but also in understanding wave patterns, oscillations, and more.
When faced with arguments like \(\frac{17\pi}{4}\), which may seem complex, we can utilize this period to simplify the expression.
By reducing the argument to a value within the initial cycle, between 0 and \(2\pi\), we can easily evaluate the trigonometric function.
This approach helps in understanding and calculating trigonometric values without needing to go beyond the first cycle.

Cosine Function

The cosine function is a fundamental trigonometric function that arises frequently in mathematics, especially in topics dealing with angles, waves, and rotations.
In its most basic form, the cosine of an angle \(\theta\) is the x-coordinate of the point on the unit circle at angle \(\theta\) from the positive x-axis.
The general form of the cosine function is \(f(\theta) = \cos \theta\), which produces a smooth, continuous curve oscillating between 1 and -1.

• Cosine functions are periodic, with a period of \(2\pi\).
• They are symmetric about the y-axis, making them an even function.
• This symmetry implies \(\cos(-\theta) = \cos(\theta)\).

These properties make the cosine function easy to analyze and understand.
For an angle such as \(\theta = \frac{17\pi}{4}\), simplifying it involves harnessing the periodic nature of the function, as discussed earlier.

Unit Circle

The unit circle is a powerful tool in understanding trigonometric functions, particularly the cosine function.
It's a circle with a radius of 1 centered at the origin of the coordinate plane.
Each point on the unit circle corresponds to an angle, \(\theta\), measured from the positive x-axis, and the coordinates of this point give the cosine and sine of that angle.

• The x-coordinate of a point on the unit circle is \(\cos(\theta)\).
• The y-coordinate is \(\sin(\theta)\).
• For \(\theta = \pi/4\), a common angle, we know \(\cos(\pi/4) = \sqrt{2}/2\).

The unit circle provides a geometric way to determine the values of trigonometric functions, including those beyond one full revolution by considering equivalent angles.
Thus, it is incredibly helpful for visualizing and understanding how trigonometric functions like cosine behave.