Problem 43 - A Graphical Approach to Precalculus with Limits

Question

Write each expression in terms of its co-function. $$\cos \frac{\pi}{5}$$

Step-by-Step Solution

Answer

\(\cos\left(\frac{\pi}{5}\right) = \sin\left(\frac{3\pi}{10}\right)\)

Step 1: Identify the Co-Function Identity

The co-function identity related to cosine is \[\cos(x) = \sin\left(\frac{\pi}{2} - x\right)\]Our task is to express \(\cos\left(\frac{\pi}{5}\right)\) using its co-function, sine.

Step 2: Apply the Identity

To express \(\cos\left(\frac{\pi}{5}\right)\) as a sine function, substitute \(x = \frac{\pi}{5}\) into the co-function identity:\[\cos\left(\frac{\pi}{5}\right) = \sin\left(\frac{\pi}{2} - \frac{\pi}{5}\right)\]

Step 3: Simplify the Expression

Simplify the expression inside the sine function:\[\frac{\pi}{2} - \frac{\pi}{5} = \frac{5\pi}{10} - \frac{2\pi}{10} = \frac{3\pi}{10}\]Therefore, the expression becomes:\[\cos\left(\frac{\pi}{5}\right) = \sin\left(\frac{3\pi}{10}\right)\]

Key Concepts

Co-function Identities

Co-function identities are unique relationships in trigonometry that connect certain trigonometric functions with each other. You can think of them as "brother-sister" relationships between angles in a right triangle. Specifically, sine and cosine are co-functions of each other, as are tangent and cotangent, secant and cosecant.
In a right triangle, the sine of an angle is equal to the cosine of its complement, and vice versa. Mathematically, for any angle \( x \), the co-function identity for cosine can be expressed as:

• \( \cos(x) = \sin\left(\frac{\pi}{2} - x\right) \)

This tells us that cosine of an angle is the same as sine of its complementary angle, because in a right triangle, the two acute angles always add up to \( \frac{\pi}{2} \).
Understanding these identities allows us to convert one trigonometric function into another, which can be incredibly helpful in solving trigonometric problems. Co-function identities provide a way to simplify expressions and make calculations easier.

Cosine Function

The cosine function is one of the three main functions in trigonometry, originating from Ancient Greek geometry. It helps in determining the horizontal dimension of a right triangle, specifically the ratio of the adjacent side over the hypotenuse.
Mathematically, this is defined as:

• \( \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} \)

The cosine function is periodic with a period of \( 2\pi \), meaning its values repeat every \( 360^{\circ} \). In the context of the unit circle, the x-coordinate of any point on the circle corresponds to the cosine of the angle formed with the positive x-axis.
The cosine function is important not only in pure mathematics but also in real-world applications such as physics (wave motion), engineering, and computer graphics. Understanding cosine helps you measure phases in oscillations and waves.

Sine Function

The sine function is another fundamental aspect of trigonometry, commonly used in the study of triangles and circles. In terms of a right-angled triangle, sine represents the ratio of the length of the opposite side to the hypotenuse.
Mathematically, sine is expressed as:

• \( \sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} \)

Just like cosine, the sine function is periodic, repeating every \( 2\pi \), and is essential in defining the y-axis coordinate in the unit circle for any given angle.
Sine has many applications, from describing oscillatory motion in physics and engineering to playing a critical role in Fourier transforms, which are used in signal processing. Understanding the sine function, and how it relates to cosine through co-function identities, strengthens one's ability to solve complex trigonometric problems with ease.