Problem 19 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) express the third part in terms of the first two. $$\beta, b ; a$$

Step-by-Step Solution

Answer

Express \(a\) as \(a = \frac{b}{\tan\beta}\).

Step 1: Understand the Given Triangle Parts

We are dealing with a right triangle \(\triangle ABC\), where \(\gamma = 90^\circ\). The problem gives us angle \(\beta\), side \(b\), and asks for \(a\) in terms of \(\beta\) and \(b\). Since \(\gamma\) is the right angle, \(\beta\) is one of the other angles.

Step 2: Recall Trigonometric Relationships in Right Triangles

In a right triangle, the tangent of an angle \(\beta\) is the ratio of the opposite side to the adjacent side. For \(\triangle ABC\), \(\tan\beta = \frac{b}{a}\).

Step 3: Solve for the Unknown Side

Rearrange the equation from Step 2 to solve for \(a\).\[ a = \frac{b}{\tan\beta} \]

Key Concepts

Right Triangle

A right triangle is a type of triangle that features one angle measuring exactly 90 degrees. This 90-degree angle is known as the right angle and is typically denoted by the Greek letter \( \gamma \), as seen in the triangle \( \triangle ABC \). The presence of this right angle simplifies calculations and allows for the use of specific trigonometric relationships.

Right triangles consist of three main parts:

• The hypotenuse: It is the longest side and opposite the right angle.
• Two legs: These sides form the right angle together.

In \( \triangle ABC \), given that \( \gamma = 90^\circ \), angles \( \alpha \) and \( \beta \) are the acute angles, and the sides are often labeled to reflect relevance to these angles. For example, side \( b \) may be opposite \( \beta \), while side \( a \) is adjacent to \( \beta \).

Understanding these basics is critical as they are foundational for solving problems using trigonometric ratios.

Tangent Function

The tangent function is a key trigonometric function used primarily in right triangles. It relates an angle of a triangle to the lengths of two specific sides. The tangent of an angle, notated as \( \tan \theta \), is calculated by dividing the length of the side opposite the angle by the length of the side adjacent to the angle.

In the context of right triangles, this relationship can be written as:\[ \tan \beta = \frac{\text{opposite}}{\text{adjacent}} \]For \( \triangle ABC \) with \( \beta \) as the angle of interest, the side opposite is \( b \) and adjacent is \( a \), hence the equation:\[ \tan \beta = \frac{b}{a} \]

This equation is fundamental to solving for missing sides and angles in right triangles. Notably, if you know \( \beta \) and side \( b \), \( a \) can be determined by rearranging the formula to:\[ a = \frac{b}{\tan \beta} \]

This ability to express one side using another within the context of a right triangle showcases the practical application of the tangent function.

Trigonometric Ratios

In right triangles, trigonometric ratios are mathematical tools that allow us to relate the angles of a triangle to the lengths of its sides. There are three primary trigonometric ratios: sine, cosine, and tangent, each formed by different combinations of side lengths.

• Sine (\( \sin \)): The ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( \cos \)): The ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\( \tan \)): The ratio of the length of the opposite side to the adjacent side.

These ratios are pivotal when solving for unknown sides or angles in a right triangle. They serve as the basis for trigonometric relationships. As seen in the original exercise for \( \triangle ABC \), the tangent ratio was particularly useful in expressing side \( a \) in terms of \( b \) and \( \beta \).

Each trigonometric ratio provides a different relationship, making them flexible and powerful tools in various mathematical and real-world applications, such as architecture, physics, and even computer graphics.