Problem 32 - Precalculus : Building Concepts and Connections

Question

Evaluate each trigonometric function without the use of a calculator. $$\cos (\arccos (0.8))$$

Step-by-Step Solution

Answer

From above analysis, the solution to \(\cos(\arccos(0.8))\) is 0.8.

Step 1: Understanding of Inverse Trigonometric Functions

First of all, we recognize that \(\arccos(0.8))\) is an inverse trigonometric function and its output gives an angle such that the cosine of this angle returns the original number - in this case, 0.8. Therefore, any real number within the domain of cosine, i.e., [0, \(\pi\)], is a possible output of \(\arccos(x)\).

Step 2: Applying the Inverse Trigonometric Function

In this step, we apply the inverse trigonometric function \(\arccos(0.8))\). This will give us an angle. Particular to the definition, this angle, let's denote it as \(A\), will satisfy \(\cos(A) = 0.8\). So, we can say that \(A = \arccos(0.8)\).

Step 3: Applying the Cosine function

Now that we have our angle \(A\), we do \(\cos(A)\). However, we've already defined \(A\) such that \(\cos(A) = 0.8\). Therefore, \(\cos(\arccos(0.8) = 0.8\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential to unlocking angles when given a trigonometric value. These functions essentially reverse the roles of trigonometric functions. For instance, while the cosine function takes an angle and gives us a ratio, the inverse cosine, or arccosine, takes a ratio and provides an angle.
Inverse trigonometric functions include:

• Arccosine (\( \arccos \))
• Arcsine (\( \arcsin \))
• Arctangent (\( \arctan \))

Each function provides a specific angle that correlates with a given trigonometric ratio. The primary role of these functions is to determine angles in different geometric situations. This is especially useful in various applications, such as when solving triangle problems and analyzing wave functions.

Cosine

The cosine function is commonly used to determine the x-coordinate of a point on a unit circle at a given angle. It is a trigonometric function and part of the foundation for understanding trigonometry. The cosine of an angle is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
Some features of the cosine function:

• Periodicity: Repeats its values every \( 2\pi \) radians.
• Range: Cosine values are between -1 and 1, inclusive.
• Domain: Typically, all real numbers. But when associated with inverse functions, it is often restricted.

The cosine function's formula is written as \( \cos(\theta) \). This function is crucial when finding the lengths of sides in a triangle or the x-component of vectors. Its precise nature and simplicity make it an indispensable tool in both pure and applied mathematics.

Arccosine

Arccosine is the inverse of the cosine function. Its primary purpose is to find the angle that corresponds to a given cosine value. In other words, if you know the cosine of an angle, \( \arccos \) helps determine that angle.
Characteristics of the arccosine function include:

• Domain: The range of values for input is from -1 to 1.
• Range: Output angles are restricted between 0 and \( \pi \) radians, capturing all possible scenarios of the cosine values crossing from -1 to 1.
• Utilization: Commonly used in locating angles in computations where the cosine is known.

Understanding arccosine is key to unraveling problems where reverse calculation from trigonometric values to angles is needed. This is particularly beneficial in geometry and certain branches of physics, where angle calculations are crucial to problem-solving.