Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write an algebraic expression that is equivalent to the expression. (Hint: Sketch a right triangle, as demonstrated in Example 8.) \(\tan \left(\arccos \frac{x}{5}\right)\)

Step-by-Step Solution

Answer

\(\tan \left(\arccos \frac{x}{5}\right) = \frac{\sqrt{5^2 - x^2}}{x}\)

Step 1: Draw a Right Triangle

Visualize the problem by drawing a right triangle. Label one of the angle as \(\theta\). This angle helps to denoting the arccos function. Given that \(\cos \theta = \frac{x}{5}\), label the adjacent side as 'x' and the hypotenuse as '5'. Using the Pythagorean theorem, you can find the value of the opposite side, which is \(\sqrt{5^2 - x^2}\).

Step 2: Use the Definition of Tangent

The function we are asked to express in a simpler form is a tangent function. The definition of tangent in terms of a right triangle is \(\tan \theta = \frac{opposite}{adjacent}\). So substitute 'opposite' and 'adjacent' with their corresponding lengths as identified from the triangle in Step 1. Thus \(\tan \theta = \frac{\sqrt{5^2 - x^2}}{x}\)

Step 3: Substitute Theta

Notice that \(\theta\) is actually the arccos function in the given problem. So, replace \(\theta\) with \(\arccos \frac{x}{5}\) in the equation from Step 2. So, \(\tan \left(\arccos \frac{x}{5}\right) = \frac{\sqrt{5^2 - x^2}}{x}\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle that has one angle equal to 90 degrees. This angle is called the right angle and forms the basis for a number of mathematical concepts and operations, including trigonometric functions.
In a right triangle, there are three sides: the opposite side, adjacent side, and the hypotenuse. The hypotenuse is the longest side, positioned opposite the right angle. The other two sides are referred to based on their relative position to one of the angles of interest. When an angle (other than the right angle) is specified:

• The side opposite this angle is called the opposite side.
• The side next to this angle (but not the hypotenuse) is called the adjacent side.

Understanding the layout of a right triangle is crucial when working with trigonometric functions, as these functions are often defined using the relationships between these sides.

Tangent Function

The tangent function (\( an\)) is one of the basic trigonometric functions used in mathematics. In the context of a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
Mathematically, this is expressed as:

• \( an \theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}}\)

In problems involving trigonometric identities and angles, the tangent function can be used to find missing side lengths or angles when certain information is known. It is particularly useful when dealing with angles in right triangles, as seen in the exercise where we related \(\tan(\arccos(\frac{x}{5}))\) to the other sides of the triangle.
By understanding how to represent the tangent function in terms of side lengths, you can simplify complex trigonometric expressions, which is essential in algebra and geometry.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, particularly within the context of right triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
This can be expressed as follows:

• \(a^2 + b^2 = c^2\)

In this formula, \(c\) is the length of the hypotenuse, while \(a\) and \(b\) are the lengths of the other two sides. This theorem is particularly powerful because it allows one to determine the length of any side of a right triangle if the lengths of the other two sides are known.
In the step-by-step solution presented for the exercise, we used the Pythagorean theorem to find the length of the opposite side. Given the adjacent side as \(x\) and the hypotenuse as \(5\), substituting into the theorem gives \(\sqrt{5^2 - x^2}\) as the length of the opposite side.
This powerful connection between the sides of right triangles is pivotal for solving problems involving trigonometric ratios, as it enables conversion between different forms of trigonometric expressions.