Problem 8 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) find the exact values of the remaining parts. $$b=7 \sqrt{2}, \quad c=14$$

Step-by-Step Solution

Answer

\(a = 7\sqrt{2}\), \(\alpha = 45^{\circ}\), \(\beta = 45^{\circ}\).

Step 1: Understand the Triangle Type

The given triangle is a right triangle with one angle, \( \gamma = 90^{\circ} \). This implies that side \( c \) is the hypotenuse of the triangle.

Step 2: Identify the Known Sides

From the problem, side \( b = 7 \sqrt{2} \) and the hypotenuse \( c = 14 \). We need to find the length of side \( a \) and the angles \( \alpha \) and \( \beta \).

Step 3: Use Pythagoras Theorem

According to the Pythagorean Theorem, for any right triangle:\[ a^2 + b^2 = c^2 \]Plug in the known values:\[ a^2 + (7\sqrt{2})^2 = 14^2 \]

Step 4: Simplify and Solve for \( a \)

Calculate \( (7\sqrt{2})^2 = 49 \times 2 = 98 \) and \( 14^2 = 196 \).Substitute these values into the equation:\[ a^2 + 98 = 196 \]\[ a^2 = 196 - 98 \]\[ a^2 = 98 \]Taking the square root of both sides, \[ a = \sqrt{98} = 7\sqrt{2} \].

Step 5: Calculate Angle \( \alpha \)

Since \( \beta = \arcsin\left(\frac{b}{c}\right) = \arcsin\left(\frac{7\sqrt{2}}{14}\right) = \arcsin\left(\frac{1}{2}\right) \). Thus, \( \beta = 45^{\circ} \).

Step 6: Calculate Angle \( \beta \)

Given that in a right triangle, \( \alpha + \beta = 90^{\circ} \), therefore:\[ \alpha = 90^{\circ} - 45^{\circ} = 45^{\circ} \]

Step 7: Compile the Results

The unknown parts of the triangle are: side \( a = 7\sqrt{2} \), angle \( \alpha = 45^{\circ} \), and angle \( \beta = 45^{\circ} \).

Key Concepts

Understanding Right Triangles

A right triangle is a special type of triangle where one angle is precisely 90 degrees, referred to as a right angle. This right angle distinctly separates the triangle into two legs and one hypotenuse. The hypotenuse is the longest side, always situated opposite the right angle.
In the given exercise, triangle ABC is a right triangle with angle \( \gamma = 90^{\circ} \). For this triangle:

• Side \( c \) is the hypotenuse.
• Sides \( a \) and \( b \) are the two legs.

This understanding is pivotal, as it forms the basis for applying various trigonometry rules and calculations to solve the triangle.

Introduction to Trigonometry

Trigonometry is a branch of mathematics that focuses on the relationships between the angles and sides of triangles. It plays a crucial role in calculating unknown parts of a triangle, especially right triangles. In this context, trigonometric ratios such as sine, cosine, and tangent are used to find missing angles and side lengths.
For example, sine is particularly useful in right triangles as it relates an angle to the opposite side over the hypotenuse. This offers a powerful tool for determining unknown values when you know some parts of the triangle.

Calculating Angles in a Triangle

Angle calculation in a right triangle involves using trigonometry and the fundamental properties of triangles. One essential property is that the sum of all interior angles of a triangle is always \( 180^{\circ} \).
Given \( \gamma \), the right angle, we can find the other two angles because:

• \( \alpha + \beta + 90^{\circ} = 180^{\circ} \)
• Solving for \( \alpha \) and \( \beta \), we find they both equal \( 45^{\circ} \), as each is complementary to the right angle.

Trigonometric functions help verify these angles by using known side lengths. For example, using the sine function, \( \beta = \arcsin(\frac{b}{c}) = 45^{\circ} \), demonstrating the balance inherent in the triangle.

Exploring Triangle Sides

In a right triangle, each side has specific relationships defined by the Pythagorean Theorem, which states \( a^2 + b^2 = c^2 \). This theorem is the key to solving unknown side lengths.
In the example, knowing \( b = 7\sqrt{2} \) and \( c = 14 \), we can solve for side \( a \):

• Use \( a^2 + (7\sqrt{2})^2 = 14^2 \) to establish the equation.
• Simplify to find \( a = 7\sqrt{2} \), confirming both legs are equal due to the angles, making the triangle isosceles.

Such calculations allow for precise understanding and solving of geometric problems.

The Role of Mathematics Education

Mathematics education, especially geometric and trigonometric understanding, is fundamental in fostering problem-solving and analytical skills. Through exercises like this, students learn to navigate and manipulate mathematical concepts and algorithms.
The process of learning through real-world examples ensures students grasp the practical applications of abstract theorems. It prepares them not only for academic pursuits but for everyday problem-solving:

• Assigning real-world relevance to math problems.
• Cultivating logical thinking and precision.

Ultimately, math education empowers students by building a framework for innovative and systemic thinking.