Problem 24 - Calculus

Question

Calculate the quantity without using the the trigonometric functions on your calculator. You are given that \(\sin (\pi / 12)=0.259\) and \(\cos (\pi / 5)=0.809 .\) You may want to draw a picture showing the angles involved and check your answer on a calculator. $$\cos \left(-\frac{\pi}{5}\right)$$

Step-by-Step Solution

Answer

\( \cos \left(-\frac{\pi}{5}\right) = 0.809 \) using the even function property of cosine.

Step 1: Understanding the Problem

We are asked to find the value of \( \cos \left(-\frac{\pi}{5}\right) \) without using trigonometric functions on a calculator. We are given \( \sin(\pi/12) = 0.259 \) and \( \cos(\pi/5) = 0.809 \).

Step 2: Applying the Cosine Even Function Property

Recall that cosine is an even function, meaning \( \cos(-x) = \cos(x) \). Therefore, \( \cos \left(-\frac{\pi}{5}\right) = \cos \left(\frac{\pi}{5}\right) \).

Step 3: Using the Given Value

We were given that \( \cos \left(\frac{\pi}{5}\right) = 0.809 \). Thus, substituting this into our equation from Step 2, we find \( \cos \left(-\frac{\pi}{5}\right) = 0.809 \).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions, often abbreviated as \( \cos \). It is used to calculate the ratio of the adjacent side to the hypotenuse in a right-angled triangle. The cosine function takes an angle as input and outputs a value between -1 and 1.
Understanding this function is crucial for solving many problems involving angles and lengths.
The function is periodic, with a period of \(2\pi\), which means that it repeats its values every \(2\pi\) radians.

• This periodic nature allows us to efficiently handle angles larger than \(2\pi\) by reducing them.
• Cosine is often paired with the sine function, \( \sin \), which works similarly but focuses on the ratio of the opposite side to the hypotenuse.

Overall, cosine and other trigonometric functions greatly simplify calculations and problem-solving involving triangles and circles.

Even Functions

Functionality wise, an even function is symmetric with respect to the y-axis. For a function to be even, it must satisfy the condition \( f(-x) = f(x) \) for all x in its domain. This characteristic of even functions makes them particularly useful in various mathematical calculations dealing with symmetries.
The cosine function is a classic example of an even function.

• This symmetry implies that the value of the cosine function is the same for both positive and negative angles of the same magnitude.
• Knowing that \( \cos(-x) = \cos(x) \) helps simplify computations, as it was used in our exercise to determine that \( \cos\left(-\frac{\pi}{5}\right) = \cos\left(\frac{\pi}{5}\right) \).

Using the properties of even functions is a powerful tool in trigonometry and other areas of mathematics.

Angle Properties

Angles are fundamental in trigonometry and they have specific properties that govern how they behave in mathematical equations and physical phenomena. Understanding these properties helps in navigating the circular nature inherent in trigonometric problems.
An important aspect of angles is their measurement in radians or degrees, with radians being the standard unit in mathematics.

• The angle \( -\frac{\pi}{5} \), for instance, represents a negative angle, which means it is measured clockwise from the axis.
• This negative angle behavior is crucial when applying trigonometric functions, as seen when utilizing the even property of the cosine function.

Proper understanding of angle properties enhances our problem-solving abilities by providing clarity on how angles interact within various trigonometric functions and the calculations involved.