Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 9-16: 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

\(c \approx 51.48\), \(\alpha \approx 29.74^{\circ}\), \(\beta \approx 60.26^{\circ}\).

Step 1: Understand the Triangle Parts Given

We are given a right triangle \(ABC\) with \(\gamma = 90^{\circ}\). The sides provided are \(a = 25\) and \(b = 45\). \(c\) is the hypotenuse, opposite the right angle (\(\gamma\)), while \(a\) and \(b\) are the legs of the triangle.

Step 2: Calculate the Hypotenuse \(c\) Using the Pythagorean Theorem

Apply the Pythagorean theorem: \[ c^2 = a^2 + b^2 \] Substitute the given values: \[ c^2 = 25^2 + 45^2 \] \[ c^2 = 625 + 2025 \] \[ c^2 = 2650 \] Taking the square root of both sides to find \(c\): \[ c = \sqrt{2650} \approx 51.48 \].

Step 3: Calculate Angle \(\alpha\) Using Trigonometry

Use the tangent function since you have the opposite \(a\) and the adjacent \(b\): \[ \tan(\alpha) = \frac{a}{b} = \frac{25}{45} \] Find \(\alpha\) by taking the arctangent: \[ \alpha = \tan^{-1}\left(\frac{25}{45}\right) \approx 29.74^{\circ} \].

Step 4: Calculate Angle \(\beta\) Using the Triangle Angle Sum

Since the sum of angles in a triangle is \(180^{\circ}\), calculate \(\beta\) as follows: \[ \beta = 180^{\circ} - \gamma - \alpha = 180^{\circ} - 90^{\circ} - 29.74^{\circ} \approx 60.26^{\circ} \].

Key Concepts

Pythagorean theorem

The Pythagorean Theorem is one of the fundamental concepts in right triangle trigonometry. It provides a simple and precise way to relate the lengths of the sides of a right triangle. According to the theorem, the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)).
Let's see how it works:

• For any right triangle, if you know the lengths of the two legs, you can find the hypotenuse using: \[ c^2 = a^2 + b^2 \]
• In reverse, if you have a hypotenuse and one leg, you can find the other leg.

In our exercise, we had \( a = 25 \) and \( b = 45 \), which we plugged into the formula to find \( c \).
The Pythagorean Theorem isn't just limited to finding the hypotenuse; it’s also crucial in determining distances in various coordinate systems.

Trigonometric functions

Trigonometric functions form the backbone of trigonometry and are essential for relating angles to side lengths in right triangles. There are six functions, but here we use tangent because we only have the opposite and adjacent sides.
The tangent of an angle \( \alpha \) in a right triangle is defined as:\[\tan(\alpha) = \frac{\text{Opposite}}{\text{Adjacent}}\]This function helps us determine the angle when the sides are known, as seen in the exercise. The calculation for angle \( \alpha \) is based on this function:

• The given sides are \( a = 25 \) (opposite) and \( b = 45 \) (adjacent).
• Thus, \( \tan(\alpha) = \frac{25}{45} \).
• By finding the arctangent or \( \tan^{-1} \), we derive \( \alpha \approx 29.74^{\circ} \).

Trigonometric functions are indispensable in physics, engineering, and many fields requiring angle measurements.

Angle calculation

Understanding how to calculate angles in a triangle is crucial, especially when dealing with right triangles. Once one angle is known, the other non-right angle can easily be calculated using geometric properties of triangles.
Here's how angle calculation works in a right triangle:

• In any triangle, the sum of all angles is \(180^{\circ} \).
• With a right triangle, one angle is always \(90^{\circ} \).

In the given problem, once we found \( \alpha \approx 29.74^{\circ} \), calculating the third angle \( \beta \) became straightforward:

• Subtract the known angles from \(180^{\circ} \):\[ \beta = 180^{\circ} - 90^{\circ} - 29.74^{\circ} = 60.26^{\circ} \]

This process highlights a fundamental principle of geometry and showcases its elegance in solving real-world problems effortlessly.