Problem 16 - 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. $$ a=0.42, \quad c=0.68 $$

Step-by-Step Solution

Answer

\( b \approx 0.535, \beta \approx 38.66^{\circ} \).

Step 1: Understand the Problem

We are given a right triangle (triangle ABC) with a right angle at vertex C, and sides a = 0.42 and c = 0.68. We need to find the missing parts b (the other leg) and angle \( \beta \).

Step 2: Use Pythagoras' Theorem

In a right triangle, the Pythagorean theorem states \( a^2 + b^2 = c^2 \). Substitute the given values into the equation: \( (0.42)^2 + b^2 = (0.68)^2 \).

Step 3: Compute \( a^2 \) and \( c^2 \)

Calculate \( a^2 = 0.42^2 = 0.1764 \) and \( c^2 = 0.68^2 = 0.4624 \).

Step 4: Solve for \( b^2 \)

Using the Pythagorean equation, find \( b^2 = 0.4624 - 0.1764 = 0.286 \).

Step 5: Find \( b \)

To find the value of \( b \), take the square root of \( b^2 \): \( b = \sqrt{0.286} \approx 0.535 \).

Step 6: Find Angle \( \beta \) Using Trigonometry

Use the tangent function for angle \( \beta \), which is defined as \( \tan(\beta) = \frac{a}{b} \). Therefore, \( \tan(\beta) = \frac{0.42}{0.535} \approx 0.785 \). Find the angle \( \beta \) by using the arctangent function: \( \beta = \arctan(0.785) \approx 38.66^{\circ} \).

Step 7: Verify with Sum of Angles

Verify \( \beta \) by checking the sum of angles in a triangle: \( \alpha + \beta + \gamma = 180^{\circ} \). Since \( \gamma = 90^{\circ} \), \( \alpha + 38.66^{\circ} = 90^{\circ} \). Therefore, \( \alpha \approx 51.34^{\circ} \), confirming our calculations.

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry. It shows the relationship between the three sides of a right triangle. This theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In formulaic terms, it is written as:

• \[ a^2 + b^2 = c^2 \]

In the context of our exercise, we know two sides: side \(a = 0.42\) and the hypotenuse \(c = 0.68\). By substituting these values into the equation, we can solve for the missing side \(b\). This involves calculating the squares of the known sides and then using subtraction to find \(b^2\), which results in \(0.286\). To find \(b\), we simply take the square root of \(b^2\), giving us \(b \approx 0.535\). Understanding this theorem allows us to find any side of a right triangle if the other two are known.

Trigonometry

Trigonometry is the study of the relationships between angles and sides in triangles, especially right triangles. In a right triangle, the basic trigonometric functions — sine, cosine, and tangent — relate the angles of the triangle to its sides. These functions are useful tools for finding unknown measurements in triangles. For angle \( \beta \) in our exercise, we used the tangent function, which is defined as the ratio of the opposite side to the adjacent side. That is,

• \[ \tan(\beta) = \frac{\text{opposite}}{\text{adjacent}} \]

Here, the opposite side to \(\beta\) is \(a\), and the adjacent side is \(b\). By calculating \( \tan(\beta) = \frac{0.42}{0.535} \approx 0.785 \), we can then use the arctangent function to find the angle.

Angle calculation

Calculating the angles in a triangle is essential to understanding its geometry. Given the values of the sides and knowing that one angle \( \gamma = 90^{\circ} \) in a right triangle, the other two angles \(\alpha\) and \(\beta\) add up to \(90^{\circ}\). In this exercise, to find angle \(\beta\), we used the arctangent function. This function, denoted as \(\arctan\), is useful for finding an angle when its tangent is known.

• \[ \beta = \arctan(0.785) \approx 38.66^{\circ} \]

To ensure our solution made sense, we checked that \(\alpha + \beta = 90^{\circ}\). Since we found \(\beta \approx 38.66^{\circ}\), \(\alpha\) should then be approximately \(51.34^{\circ}\), perfectly complimenting \(\beta\) to fit the triangle's angle sum rule.

Triangle sides

Understanding the sides of a triangle is crucial for solving geometric problems. In a right triangle, there are three sides to consider: two legs and the hypotenuse. Usually, the hypotenuse is the longest side, opposite the right angle. The parts we need to solve for may vary based on what is given.In this exercise, side \(a\) is one leg, with a length of \(0.42\), and \(c\) is the hypotenuse at \(0.68\). We needed to find the length of the other leg, \(b\). By using the Pythagorean theorem, we found \(b\approx 0.535\). Parsing through triangle side lengths using these methods ensures precise calculations, anchoring further explorations into angle measures and triangle properties.