Question

Use the fact that the trigonometric functions are periodic to find the exact value of each expression. Do not use a calculator. $$ \cos \frac{37 \pi}{6} $$

Short answer

The exact value is \(\frac{\sqrt{3}}{2}\).

Step-by-step solution

Determine the period of the cosine function

The cosine function is periodic with a period of \(2\pi\). This means that \(\cos(x) = \cos(x + 2k\pi)\) for any integer \(k\).

Reduce the angle within one period

Given the angle \(\frac{37\pi}{6}\), find an equivalent angle between \(0\) and \(2\pi\) by subtracting multiples of \(2\pi\). First, express \(2\pi\) with a common denominator: \(2\pi = \frac{12\pi}{6}\).

Simplify the angle

Subtract \(2\pi\) from \(\frac{37\pi}{6}\) multiple times until the resulting angle lies between \(0\) and \(2\pi\). Calculate: \(\frac{37\pi}{6} - 2\pi = \frac{37\pi}{6} - \frac{12\pi}{6} = \frac{25\pi}{6}\).

Continue reducing the angle

Subtracting again: \(\frac{25\pi}{6} - 2\pi = \frac{25\pi}{6} - \frac{12\pi}{6} = \frac{13\pi}{6}\).

One more reduction

Subtract another \(2\pi\): \(\frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}\).

Evaluate the cosine function

Now that the angle is reduced within one period, \(\cos \left(\frac{37\pi}{6}\right) = \cos \left(\frac{\pi}{6}\right) \). The exact value of \(\cos \left(\frac{\pi}{6}\right) \) is \(\frac{\sqrt{3}}{2}\).

Key concepts

Periodic Functions

Periodic functions are functions that repeat their values in regular intervals or periods. This means if you shift the input value by a certain period, the output value remains the same. Examples of periodic functions include sine, cosine, and tangent. The most prominent feature of a periodic function is its period. For instance, the cosine function has a period of \(2\pi\). This means that for any angle \(x\), \(\cos(x) = \cos(x + 2k\pi)\) where \(k\) is any integer.
Understanding the periodicity of functions is crucial in simplifying trigonometric expressions. In our example, we take advantage of the cosine function's periodicity to reduce the given angle, \(\cos \left(\frac{37\pi}{6}\right)\), to a simpler angle within one period. This process avoids the need for a calculator and builds a strong foundational understanding of trigonometric principles.

Cosine Function

The cosine function is one of the key functions in trigonometry, often represented as \(\cos(x)\). It is defined as the adjacent side over the hypotenuse in a right-angled triangle or as the x-coordinate of a point on the unit circle corresponding to an angle \(x\). The cosine function oscillates between -1 and 1 and has a period of \(2\pi\), which makes it a periodic function.
In the given exercise, we are dealing with the cosine of \(\frac{37\pi}{6}\), which is a large angle. The periodic nature of the cosine function allows us to reduce this angle to an equivalent angle within the range of \(0\) to \(2\pi\). By reducing \(\frac{37\pi}{6}\) progressively by \(2\pi\) multiples, we simplify it eventually to \(\frac{\pi}{6}\). This reduced angle makes it easier to evaluate the cosine, giving us \(\cos \left(\frac{37\pi}{6}\right) = \cos \left(\frac{\pi}{6}\right)\). Knowing standard values of cosine, we find that the exact value is \(\frac{\sqrt{3}}{2}\).

Angle Reduction

Angle reduction is a technique used to simplify trigonometric expressions by reducing a given angle to its smallest positive equivalent angle within the fundamental period of the function. This process often involves subtracting or adding full cycles of the period (for cosine and sine, this period is \(2\pi\)) until the angle falls within the desired range.
For the given exercise, we start with \(\frac{37\pi}{6}\). Since the period of the cosine function is \(2\pi\), we keep subtracting \(2\pi\) (or \(\frac{12\pi}{6}\) to maintain common denominators) until the angle lies between \(0\) and \(2\pi\).
Here's how it works step-by-step:

• \(\frac{37\pi}{6} - 2\pi = \frac{37\pi}{6} - \frac{12\pi}{6} = \frac{25\pi}{6}\)
• \(\frac{25\pi}{6} - 2\pi = \frac{25\pi}{6} - \frac{12\pi}{6} = \frac{13\pi}{6}\)
• \(\frac{13\pi}{6} - 2\pi = \frac{13\pi}{6} - \frac{12\pi}{6} = \frac{\pi}{6}\)

Finally, after reducing \(\frac{37\pi}{6}\) to \(\frac{\pi}{6}\), we evaluate \(\cos \left(\frac{\pi}{6}\right)\), which gives us \(\frac{\sqrt{3}}{2}\). This process not only simplifies the computation but also helps in understanding the cyclic nature of trigonometric functions.