Problem 60 - Algebra 2

Question

In \(\triangle A B C, \angle B\) is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth. $$ a=4.1, b=9.4 $$

Step-by-Step Solution

Answer

The hypotenuse of the triangle is approximately 10.1 units long. The measure of \(\angle A\) is approximately 23.7 degrees, and the measure of \(\angle C\) is approximately 66.3 degrees.

Step 1: Determine the Length of the Hypotenuse (c)

Since \(\triangle ABC\) is a right triangle and \(\angle B\) is the right angle, side 'c' is the hypotenuse. Use the Pythagorean theorem (\(a^2 + b^2 = c^2\)) to find its length. Substitute the given values \(a = 4.1\) and \(b = 9.4\) into the formula to get \(c^2 = (4.1)^2 + (9.4)^2\). Calculate the value under the square root first, then take the square root of the result to get 'c'.

Step 2: Calculate Angles 'A' and 'C'

Use the tangent function to find \(\angle A\) due to the relation \(tan(A) = \frac{a}{b}\), where 'a' is the opposite side to \(\angle A\) and 'b' is the adjacent side to \(\angle A\). Substitute the given lengths into the formula and calculate to get 'A'. Angle 'C' can be found by the fact that the sum of the angles in a triangle is 180 degrees. So \(\angle C = 180 - 90 - \angle A\) since \(\angle B\) is a right angle. Calculate 'C' to complete the triangle.

Key Concepts

Right Triangle

A right triangle, as mentioned in the exercise, is a triangle that includes a 90-degree angle, often labeled as the "right angle." In triangle notation, the right angle is typically marked with a small square in one of the corners. This angle is always present in right triangles, distinguishing them from other types of triangles.
This specific feature allows the application of the Pythagorean theorem, a crucial method for calculating the sides of the triangle when some measurements are known.
Regular triangles have angle measures that sum up to 180 degrees, making it easier to find missing angles when working with a right triangle.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. For right triangles, it becomes simpler due to the fixed angle provided by the right angle.
The key functions used in trigonometry are sine (sin), cosine (cos), and tangent (tan). In this exercise, the tangent function is used to find one of the unknown angles as it relates the opposite side to the adjacent side of the given angle. This relationship is defined as:

• tan(\(A\)) = \(\frac{a}{b}\)

where 'a' and 'b' are opposite and adjacent sides respectively. By knowing these sides, one can find the angle measure using the inverse of the tangent function, allowing us to solve for angles effectively.

Angle Calculation

To calculate the angles in a right triangle after determining the lengths of all sides, you begin by using the tangent function as outlined in trigonometry.
First, calculate \(\angle A\) using tan:\(A = \frac{a}{b}\). Insert the values for the sides, and use a calculator to find the arctangent (inverse tangent), which will yield \(\angle A\).
Once \(\angle A\) is found, remember that the sum of angles in any triangle is 180 degrees. Thus, in a right triangle, the relation can be expressed as:

• \(\angle C = 180^\circ - 90^\circ - \angle A\)

Use this formula to determine \(\angle C\), ensuring that all angles add up correctly in the triangle.