Problem 14 - A Graphical Approach to Precalculus with Limits

Question

Solve each right triangle. In each case, \(C=90^{\circ} .\) If angle information is given in degrees and minutes, give answers in the same way. If given in decimal degrees, do likewise in answers. When two sides are given, give angles in degrees and minutes. \(b=15\) feet; \(c=17\) feet

Step-by-Step Solution

Answer

With side lengths \(a = 8\) feet, \(b = 15\) feet, \(c = 17\) feet, the angles are \(A = 61^\circ\;56'\) and \(B = 28^\circ\;4'\).

Step 1: Identify the Given Information

We have a right triangle with known sides: \(b = 15\) feet and \(c = 17\) feet. Since \(C\) is the right angle, \(C = 90^\circ\), \(b\) is the opposite side to angle \(A\), and \(c\) is the hypotenuse.

Step 2: Calculate Side 'a' Using Pythagorean Theorem

The Pythagorean Theorem states that \(a^2 + b^2 = c^2\). Substitute the known values: \(a^2 + 15^2 = 17^2\). This simplifies to \(a^2 + 225 = 289\). Solving for \(a^2\), we get \(a^2 = 64\). Taking the square root gives \(a = 8\) feet.

Step 3: Calculate Angle 'A' Using Sine Relation

Since we know \(b\) and \(c\), use the sine function: \(\sin(A) = \frac{b}{c}\). Substitute the values: \(\sin(A) = \frac{15}{17}\). Calculate \(A\) using a calculator to find \(A \approx 61.93^\circ\). Convert decimal degrees to degrees and minutes: \(61.93^\circ = 61^\circ\;55.8'\). Rounding gives \(A \approx 61^\circ\;56'\).

Step 4: Calculate Angle 'B' Using Angle Sum Property

In a triangle, the sum of angles is \(180^\circ\). Thus, \(A + B + C = 180^\circ\). Knowing \(A = 61^\circ\;56'\) and \(C = 90^\circ\), we find \(B = 180^\circ - 90^\circ - 61^\circ\;56' = 28^\circ\;4'\).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This right angle is a key characteristic that defines it. In every right triangle, the side opposite to the right angle is known as the hypotenuse, which is the longest side of the triangle. The other two sides are referred to as the 'legs'. Understanding the structure of a right triangle is crucial when solving problems related to it. In these triangles:

• The side opposite the right angle, the hypotenuse, is of great importance.
• The legs are adjacent to the right angle and help define the other angles and positions within the triangle.
• Angles other than the right angle are acute, meaning they are less than 90 degrees.

Right triangles are fundamental to trigonometry and are commonly used as a basis for solving many geometric problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle used in geometry, particularly with right triangles. It allows us to relate the lengths of sides in a right triangle. The theorem is expressed as:\[a^2 + b^2 = c^2\]Here:

• \(a\) and \(b\) represent the lengths of the legs.
• \(c\) represents the hypotenuse.

This formula is only applicable to right triangles and is a powerful tool for calculating the unknown side when the other two are given. For instance, if you know the hypotenuse and one leg, you can solve for the other leg. In our exercise, we found that when given \(b = 15\) and \(c = 17\), solving for \(a\) gives \(a = 8\), demonstrating the effectiveness of this theorem.Employing the Pythagorean Theorem is essential for many applications beyond just planar geometry, including physics and engineering.

Sine Function

In trigonometry, the sine function is used to relate the angles and sides of a triangle. Specifically, in right triangles, it helps find angles when certain sides are known.The sine of an angle \(A\) in a right triangle is defined as the ratio of the length of the side opposite to angle \(A\) to the hypotenuse. The formula is:\[\sin(A) = \frac{b}{c}\]In our context:

• \(b\) is the length of the side opposite the angle \(A\).
• \(c\) is the hypotenuse.

By substituting known values, like \(b = 15\) and \(c = 17\), we applied this formula and determined \(A\) using a calculator.The sine function is pivotal in trigonometry not only for solving triangle problems but also for analyzing waves and circular motion in more advanced mathematics and physics.

Angle Sum Property

The angle sum property is a fundamental concept stating that the sum of the internal angles in any triangle is always 180 degrees. This is particularly useful in right triangles, where one of the angles is already defined as 90 degrees. Therefore, in a right triangle:

• The sum of the other two angles must be 90 degrees.
• If one angle is known, you can easily find the other by subtracting the known acute angle from 90 degrees.

For example, in our problem, after computing angle \(A = 61^\circ\;56'\), we find \(B\) by subtracting \(A\) from 90 degrees. This approach simplifies solving triangles and is one of the key strategies for deducing unknown angles in triangle geometry. Understanding this property is critical for anyone studying or applying trigonometry.