Problem 35 - Algebra 2

Question

In \(\triangle A B C, \angle C\) is a right angle. Two measures are given. Find the remaining sides and angles. Round your answers to the nearest tenth. \(b=8, c=17\)

Step-by-Step Solution

Answer

The length of side a is 15 units, the measure of angle A is 28.1 degrees, and the measure of angle B is 61.9 degrees.

Step 1: Find the Length of Side a

In a right triangle, the Pythagorean Theorem states that \(a^{2} = c^{2} - b^{2}\), where c is the hypotenuse, and a and b are the other two sides. Plugging in the given values, we get \(a^{2} = 17^{2} - 8^{2} = 225\). Taking the square root of both sides, we find that \(a = 15\).

Step 2: Find the Measure of Angle A

In a right triangle, the angle measures can be found using the tangent function. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, \(\tan A = \frac{b}{a} = \frac{8}{15}\). Using a calculator to find the arctan of this value, we find that \(A = 28.1\) degrees.

Step 3: Find the Measure of Angle B

In a right triangle, the measure of the remaining angle can be found by subtracting the other two angle measures from 180 degrees (since the sum of the measures of the angles in any triangle is 180 degrees). In this case, \(B = 90 - 28.1 = 61.9\) degrees.

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles measures exactly 90 degrees. This unique feature allows us to use several mathematical rules and functions to determine unknown sides and angles. In a right triangle, the side opposite the right angle is called the hypotenuse. This is always the longest side. The other two sides are known as the adjacent and opposite, based on the angle we are examining. Right triangles are foundational in geometry and trigonometry, as they allow us to explore relationships between angles and sides using the Pythagorean Theorem and trigonometric functions.

Trigonometric Functions

Trigonometric functions are mathematical tools that relate the angles of a triangle to the lengths of its sides. In a right triangle, the main trigonometric functions are sine, cosine, and tangent.

• Sine (\( ext{sin}\)) is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( ext{cos}\)) is the ratio of the adjacent side to the hypotenuse.
• Tangent (\( ext{tan}\)) is the ratio of the opposite side to the adjacent side.

These functions are essential in solving triangles when you know certain sides or angles, and they help in finding unknown parts of the triangle.

Triangle Angles

In any triangle, the sum of the internal angles is always 180 degrees. This principle holds true for right triangles as well. In a right triangle, since one angle is always 90 degrees, the other two angles must add up to 90 degrees. Knowing this allows you to find any missing angle if the other is known. Simply subtract the known angles from 180 degrees to find the missing one. This concept is fundamental in solving problems involving triangle angles, especially when working with trigonometric functions.

Arctan Function

The arctan, or inverse tangent function, is a powerful tool for finding an angle when you know the tangent ratio. In our example, when we found \( an A = \frac{8}{15}\), we used the arctan function to determine that \(A = 28.1\) degrees. This function reverses what tangent does, allowing us to go from the ratio back to the angle measurement. It's especially useful in right triangle problems when you have the opposite and adjacent sides and need to find the angle between them. Using a calculator, the arctan function quickly gives the angle to a desired precision.