Problem 14 - 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. $$ a=31, \quad b=9.0 $$

Step-by-Step Solution

Answer

The remaining side length is approximately 32.28, \(\alpha\) is 16.21°, and \(\beta\) is 73.79°.

Step 1: Identify Given Parts

In this right triangle, angle \(\gamma\) is 90 degrees, meaning it is a right triangle. We are also given side \(a = 31\) and side \(b = 9.0\). Our goal is to find the remaining side and angles.

Step 2: Use Pythagorean Theorem

To find the length of side \(c\), we apply the Pythagorean Theorem which states \(c^2 = a^2 + b^2\). Thus, we calculate \[ c = \sqrt{31^2 + 9^2} = \sqrt{961 + 81} = \sqrt{1042}. \] Compute \(c\) to get approximately \(32.28\).

Step 3: Find Angle \(\alpha\) Using Sine

Use the sine function, which is defined as the opposite side over the hypotenuse, to find \(\alpha\). \[ \sin(\alpha) = \frac{b}{c} = \frac{9}{32.28} \approx 0.279\] Use the inverse sine to solve for \(\alpha\): \[ \alpha = \sin^{-1}(0.279) \approx 16.21^{\circ}. \]

Step 4: Find Angle \(\beta\)

Since the sum of angles in a triangle is \(180^{\circ}\), use the equation \(\alpha + \beta + \gamma = 180^{\circ}\). Substitute \(\alpha\) and \(\gamma\): \[ 16.21 + \beta + 90 = 180 \] Hence, \(\beta\), can be found as: \[ \beta \approx 180 - 90 - 16.21 = 73.79^{\circ}. \]

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially for right triangles. It relates the lengths of the sides in a right triangle. The theorem is represented as \(c^2 = a^2 + b^2\), where \(c\) is the hypotenuse—the longest side opposite the right angle—and \(a\) and \(b\) are the other two sides. This formula helps us find the unknown side length of a right triangle if the other two side lengths are known.
In our exercise, since side \(a = 31\) and side \(b = 9.0\) are given, we can find the hypotenuse \(c\) by calculating \( \sqrt{31^2 + 9^2} \). This yields approximately \(32.28\), which is the length of side \(c\). Remember, this theorem is only applicable in right triangles!

sine function

The sine function is a well-known concept in trigonometry, especially useful for solving right triangles. In a right triangle, the sine of an angle is the ratio between the length of the side opposite the angle and the hypotenuse. Mathematically defined as \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
To find angle \( \alpha \) in the exercise, we use \( \sin(\alpha) = \frac{9}{32.28} \), providing us with a decimal value of approximately \(0.279\). This ratio effectively links the triangle's sides with its angles. Understanding this relationship is crucial, as it allows us to move seamlessly from knowing side lengths to determining unknown angles.

inverse trigonometric functions

Inverse trigonometric functions are vital when you need to find angles, especially when the trigonometric ratios are known. The inverse sine function, denoted as \( \sin^{-1} \), is used to find the angle when its sine value is given. For instance, if \( \sin(\alpha) = 0.279 \), then \( \alpha = \sin^{-1}(0.279) \), approximately leading to \(16.21^{\circ}\).
This step is particularly important for right triangles where you might have side lengths and need to find specific angle measurements. By calculating the inverse, students can effectively turn trigonometric values back into meaningful angles, which can further help solve the entire triangle problem.

angle sum property of triangles

The angle sum property is a basic principle in geometry stating that the sum of all interior angles in any triangle is \(180^{\circ}\). In the context of a right triangle, one angle is \(90^{\circ}\) by definition. Therefore, the sum of the other two angles must be \(90^{\circ}\) to meet the \(180^{\circ}\) total.
With our triangle having a right angle \(\gamma = 90^{\circ}\) and finding \(\alpha \approx 16.21^{\circ}\), we can find \(\beta\) using \(16.21 + \beta + 90 = 180\). Thus, \(\beta = 73.79^{\circ}\). This property is particularly useful when two angles are already known, leaving the third to be easily calculated. Keeping this in mind ensures the solutions are comprehensive and checks your work in complex trigonometry problems.