Problem 13 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) approximate the remaining parts. $$a=25, \quad b=45$$

Step-by-Step Solution

Answer

The hypotenuse \( c \approx 51.48 \), \( \alpha \approx 29.05^{\circ} \), and \( \beta \approx 60.95^{\circ} \).

Step 1: Understanding the Problem

You are given a right-angled triangle with one angle \( \gamma = 90^{\circ} \) and two sides, \( a = 25 \) and \( b = 45 \). We need to find the remaining side \( c \) and the other two angles \( \alpha \) and \( \beta \).

Step 2: Using the Pythagorean Theorem

The Pythagorean theorem states that in any right-angled triangle, the square of the hypotenuse \( c \) is equal to the sum of the squares of the other two sides. Thus, we calculate \( c \) using the equation:\[ c = \sqrt{a^2 + b^2} = \sqrt{25^2 + 45^2}. \]Calculate the values:\[ c = \sqrt{625 + 2025} = \sqrt{2650} \approx 51.48. \]

Step 3: Calculating Angle \( \alpha \) using Tangent

To find angle \( \alpha \), use the tangent function which relates the opposite side \( a \) to the adjacent side \( b \):\[ \tan(\alpha) = \frac{a}{b} = \frac{25}{45}. \]Calculate \( \alpha \) using inverse tangent:\[ \alpha = \tan^{-1}\left(\frac{25}{45}\right) \approx 29.05^{\circ}. \]

Step 4: Calculating Angle \( \beta \)

Since the sum of angles in any triangle is \( 180^{\circ} \) and angle \( \gamma = 90^{\circ} \), calculate \( \beta \) as follows:\[ \beta = 90^{\circ} - \alpha = 90^{\circ} - 29.05^{\circ} = 60.95^{\circ}. \]

Key Concepts

Pythagorean theorem

In a right triangle, the Pythagorean theorem is a fundamental principle. It establishes a relationship between the three sides of the triangle. Specifically, it declares that the square of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the other two sides. In mathematical terms, this is written as \( c^2 = a^2 + b^2 \).
When you're given two sides of a right triangle, you can use this theorem to find the third side. For instance, if side \( a = 25 \) and side \( b = 45 \), the hypotenuse \( c \) would be calculated like this:

• First, square both given sides: \( 25^2 = 625 \) and \( 45^2 = 2025 \).
• Next, add these squares: \( 625 + 2025 = 2650 \).
• Finally, take the square root of this sum to find \( c \): \( c = \sqrt{2650} \approx 51.48 \).

This theorem is incredibly useful in geometry and everyday applications where distance measurement is needed.

trigonometric ratios

Trigonometric ratios are a cornerstone of trigonometry, involving the comparison of the lengths of sides in a right triangle. The primary trigonometric ratios are sine, cosine, and tangent. Each of these relates a pair of a triangle's sides to one of its angles, except the right angle.
For the angle \( \alpha \) in a right triangle:

• Sine (sin): \( \sin(\alpha) = \frac{\text{opposite side}}{\text{hypotenuse}} \).
• Cosine (cos): \( \cos(\alpha) = \frac{\text{adjacent side}}{\text{hypotenuse}} \).
• Tangent (tan): \( \tan(\alpha) = \frac{\text{opposite side}}{\text{adjacent side}} \).

These ratios are useful for solving triangles, especially when given limited information, such as just one angle and one side. In the exercise, the tangent function relates side \( a = 25 \) to side \( b = 45 \), allowing us to find the angle \( \alpha \) with the formula \( \tan(\alpha) = \frac{25}{45} \).

inverse tangent function

The inverse tangent function, also known as arctangent, is crucial when you need to determine an angle in a right triangle but only have the lengths of the opposite and adjacent sides. This function is generally denoted as \( \tan^{-1} \) or arctan.
When using the inverse tangent, you start with the ratio of the opposite side to the adjacent side, and you use the function to find the angle. For example, in our triangle, we have \( \tan(\alpha) = \frac{25}{45} \). By applying the inverse tangent:

• Firstly, compute the ratio: \( \frac{25}{45} = 0.5556 \).
• Then, find \( \alpha = \tan^{-1}(0.5556) \).
• This results in \( \alpha \approx 29.05^{\circ} \).

This step is vital for solving right triangles when angles are unknown.

triangle angle sum property

The triangle angle sum property is a fundamental concept in geometry that states the sum of the internal angles of any triangle is always \( 180^{\circ} \). In a right triangle, one angle is always \( 90^{\circ} \), leaving the other two angles to sum up to \( 90^{\circ} \).
To find angle \( \beta \) in our problem, knowing that \( \alpha \approx 29.05^{\circ} \), you subtract \( \alpha \) from \( 90^{\circ} \) to maintain this required sum.

• Start by adding the known angles: \( \beta + \alpha = 90^{\circ} \).
• Solve for \( \beta \): \( \beta = 90^{\circ} - 29.05^{\circ} = 60.95^{\circ} \).

This property simplifies finding unknown angles when you already have one or more angle measures.