Problem 15 - Algebra 2

Question

Find the exact value of each function. \(\cos 5 \pi\)

Step-by-Step Solution

Answer

The exact value of \(\cos 5\pi\) is \(-1\).

Step 1: Understanding the Function

The exercise asks us to find the exact value of the cosine function at the angle \(5\pi\). The angle is given in radians.

Step 2: Determine the Angle's Position

Since cosine is a periodic function with a period of \(2\pi\), we can find an equivalent angle for \(5\pi\) by subtracting multiples of \(2\pi\) until the angle is within one full rotation (from \(0\) to \(2\pi\)). Thus, \(5\pi - 2\pi = 3\pi\). Since \(3\pi\) is more than one full rotation, subtract another \(2\pi\). So, \(3\pi - 2\pi = \pi\).

Step 3: Evaluate the Equivalent Angle

Now we need to evaluate \(\cos \pi\) because \(\cos 5\pi = \cos \pi\). The value of \(\cos \pi\) is known to be \(-1\).

Step 4: Conclusion

Thus, the exact value of \(\cos 5\pi\) is \(-1\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, alongside sine and tangent. It relates the angle of a right triangle to the lengths of its adjacent side and hypotenuse. When considering the unit circle, the cosine of an angle is the x-coordinate of the point where the terminal side of the angle intersects the circle.
For example:

• At 0 radians, \(\cos 0 = 1\).
• As the angle increases to \(\frac{\pi}{2}\) (90 degrees), \(\cos \frac{\pi}{2} = 0\).
• At \(\pi\) (180 degrees), \(\cos \pi = -1\).
• The function continues this periodic pattern around the unit circle.

The pattern of the cosine function repeats every \(2\pi\) radians. This characteristic is known as periodicity. Understanding this cyclic nature is crucial for solving problems involving angles that exceed \(2\pi\), such as \(5\pi\).

Angle in Radians

Angles can be measured in degrees or radians. The radian is a standard unit of angular measure that is used in many areas of mathematics. One complete rotation around a circle is \(2\pi\) radians, which is the same as 360 degrees.
This means:

• Half a circle (straight angle) is \(\pi\) radians.
• A right angle is \(\frac{\pi}{2}\) radians.

To convert an angle from degrees to radians, use the formula:
\[\text{Radians} = \left( \frac{\pi}{180} \right) \times \text{Degrees}\]
For example, 90 degrees can be converted to radians as follows:
\[\frac{\pi}{2} = \left( \frac{\pi}{180} \right) \times 90\]
Conversely, to convert from radians to degrees:
\[\text{Degrees} = \left( \frac{180}{\pi} \right) \times \text{Radians}\] Understanding radians is very helpful in calculus and other advanced math topics, especially when dealing with periodic properties of trigonometric functions like cosine.

Exact Value of Trigonometric Functions

The exact value of a trigonometric function refers to its precise, non-approximate measurement. To find this, particularly when dealing with the cosine function at certain angles, you don't always need a calculator.
Key angles like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) and their multiples are essential, as their cosine values are well-known:

• \( \cos 0 = 1 \)
• \( \cos \frac{\pi}{2} = 0 \)
• \( \cos \pi = -1 \)
• \( \cos \frac{3\pi}{2} = 0 \)

To find \(\cos 5\pi\), we make use of the periodic nature of cosine. \(5\pi\) is equivalent to \(\pi\) in terms of the unit circle, as shown by reducing \(5\pi - 2\pi - 2\pi = \pi\).
Thus, \(\cos 5\pi = \cos \pi = -1\).
This offers a neat resolution without complex calculations, highlighting the effectiveness of understanding trigonometric identities and the unit circle.