Problem 49 - Algebra 2

Question

Find the exact value of each function. \(\cos 25 \pi\)

Step-by-Step Solution

Answer

\(\cos(25\pi) = -1\)

Step 1: Understand the Cosine Function Periodicity

The cosine function is periodic with a period of \(2\pi\). This means \(\cos(\theta) = \cos(\theta + 2\pi k)\) for any integer \(k\). We can use this property to find an equivalent angle within one complete cycle (0 to \(2\pi\)).

Step 2: Find the Equivalent Angle

Since \(\cos\) repeats every \(2\pi\), we subtract multiples of \(2\pi\) from \(25\pi\) to find an equivalent angle within the range \([0, 2\pi)\). Calculate \(25\pi \mod 2\pi\).

Step 3: Calculate 25π mod 2π

Divide the coefficient of \(\pi\) which is 25 by 2. \(25 \div 2 = 12.5\). This means that \(25\pi = 12 \times 2\pi + 1\pi\). Thus, \(25\pi \mod 2\pi = \pi\).

Step 4: Evaluate Cosine of the Resulting Angle

Now that we have reduced \(25\pi\) to \(\pi\), we evaluate \(\cos(\pi)\). From trigonometric identities, \(\cos(\pi) = -1\).

Step 5: Conclusion

Using periodicity, we've found that the large angle \(25\pi\) is equivalent to \(\pi\) in one complete cycle. Therefore, \(\cos(25\pi) = \cos(\pi) = -1\).

Key Concepts

Periodicity of Trigonometric Functions

Trigonometric functions, like cosine, exhibit a fascinating property known as periodicity. This means they follow a repeating pattern over regular intervals. For the cosine function, this period is exactly \(2\pi\). Consequently, for any angle \(\theta\), the cosine of that angle is equal to the cosine of any angle that is \(2\pi\) plus \(\theta\). This can be written as:

• \( \cos(\theta) = \cos(\theta + 2\pi k) \) for any integer \(k\)

Understanding this is crucial because it allows us to simplify calculations. Rather than working with very large angles, we can "loop" them back into a typical cycle of \([0, 2\pi)\). For example, to find \(\cos(25\pi)\), you can apply this property because \(25\pi\) is just several full cycles beyond \(\pi\). Using the periodicity, we find that \(\cos(25\pi)\) is the same as \(\cos(\pi)\), which makes the calculations simpler.

Cosine Function

The cosine function, symbolized by \(\cos(\theta)\), is one of the primary trigonometric functions and represents the x-coordinate of a point on the unit circle at an angle \(\theta\) from the positive x-axis. Its values range between -1 and 1. Here are some basic properties:

• \( \cos(0) = 1 \)
• \( \cos(\pi/2) = 0 \)
• \( \cos(\pi) = -1 \)
• \( \cos(3\pi/2) = 0 \)
• \( \cos(2\pi) = 1 \)

These key positions highlight the cosine's pattern of beginning at 1, descending to -1, and rising back to 1 over a full \(2\pi\) cycle. Therefore, when calculating a cosine for a large angle, first reduce the angle using periodicity, and then apply these known values.

Angle Reduction

Angle reduction is a strategy used to simplify trigonometric problems by reducing a large angle to an equivalent angle within a standard range, usually \([0, 2\pi)\). For the problem \(\cos(25\pi)\), angle reduction is achieved through applying the mod operation:

• Compute \(25\pi \mod 2\pi\)
• This calculation simplifies to \(\pi\) after finding the remainder when dividing 25 by 2 (\(25 \div 2 = 12\) with a remainder \(1\))

The equivalent angle \(\pi\) then allows you to use known cosine values. Thus, \(\cos(25\pi) = -1\). Angle reduction not only makes calculations more manageable but also provides an efficient method to tackle otherwise complex trigonometric problems.