Problem 42 - Intermediate Algebra

Question

Find the missing lengths in each triangle. Give the exact answer and then an approximation to two decimal places, when appropriate. See Example \(6 .\) In a \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangle, the length of the hypotenuse is \(12 \sqrt{3}\) inches. Find the length of the leg opposite the \(30^{\circ}\) angle and the length of the leg opposite the \(60^{\circ}\) angle. Give the exact answer and then an approximation to two decimal places, when appropriate.

Step-by-Step Solution

Answer

The side opposite the 30° angle is 6√3 ≈ 10.39 inches. The side opposite the 60° angle is 18 inches.

Step 1: Understanding the 30°-60°-90° Triangle

A 30°-60°-90° triangle is a special type of right triangle where the sides are in the ratio 1:√3:2. The side opposite the 30° angle is the smallest, the side opposite the 60° angle is the intermediate, and the hypotenuse is the longest. Here, we know the hypotenuse is given as 12√3 inches.

Step 2: Finding the Length of the Smaller Leg

In a 30°-60°-90° triangle, the length of the side opposite the 30° angle (the smallest side) is half of the hypotenuse. Since the hypotenuse is 12√3 inches, the side opposite the 30° angle is \( \frac{12\sqrt{3}}{2} = 6\sqrt{3} \) inches.

Step 3: Finding the Length of the Larger Leg

The side opposite the 60° angle is \( \sqrt{3} \) times the smaller leg. We have found the smaller leg to be 6√3 inches, so the side opposite the 60° angle is \( 6\sqrt{3} \times \sqrt{3} = 18 \) inches.

Step 4: Approximating to Two Decimal Places

Convert the exact lengths to decimal form. The smaller leg is \(6\sqrt{3} \approx 10.39\) inches (since \(\sqrt{3} \approx 1.732\)). The larger leg is already in whole numbers (18.00 inches), so no further approximation is needed.

Key Concepts

Special Right Triangles

A 30°-60°-90° triangle is an example of a special right triangle. These triangles have unique angle measures and side length ratios that make calculations easier. In a 30°-60°-90° triangle, the angles are always 30 degrees, 60 degrees, and 90 degrees. Typically, these triangles have sides in a fixed ratio of 1:√3:2. This means:

• The side opposite the 30° angle is the shortest and often labeled as "1".
• The side opposite the 60° angle is √3 times the shortest side.
• The hypotenuse, the longest side, is twice the shortest side.

Understanding these ratios allows you to swiftly determine unknown side lengths when at least one side is known. This property of 30°-60°-90° triangles saves time and simplifies solving problems, as you don't need to resort to lengthy calculations each time you encounter such a triangle. Remembering the ratio 1:√3:2 can make a world of difference in approaching related trigonometry problems.

Trigonometry Ratios

Trigonometry involves ratios that can be very useful, especially when dealing with right triangles. In a 30°-60°-90° triangle, the trigonometric ratios are based on the side lengths in relation to the triangle's angles. Think of these ratios as tools that relate the angles inside a triangle to its side lengths:

• Sine (sin) of an angle is defined as the ratio of the opposite side to the hypotenuse. For 30°, sin(30°) = 1/2.
• Cosine (cos) is the ratio of the adjacent side to the hypotenuse. For 60°, cos(60°) = 1/2.
• Tangent (tan) is the ratio of the opposite side to the adjacent side. For 60°, tan(60°) = √3.

These fundamental trigonometric ratios provide an additional method for calculating the missing sides of triangles, reinforcing the consistency of the special right triangle relationships. Once you're comfortable with these ratios, you can effortlessly apply them to validate or find missing sides of given triangles.

Hypotenuse Calculation

In right triangles, the hypotenuse is the side opposite the 90° angle and is always the longest. Calculating the hypotenuse becomes straightforward with special right triangles where side ratios are consistent. For instance, in a 30°-60°-90° triangle:

• Once you know the hypotenuse, determining the other sides is straightforward based on the ratio 1:√3:2. For example, if the hypotenuse is 12√3 inches, the side opposite the 30° angle is half of that, 6√3 inches.
• For the side opposite the 60° angle, it is √3 times the shorter side, giving us 18 inches (since 6√3 × √3 = 18).

Thus, the hypotenuse plays a pivotal role in solving for unknown values, but remember, its calculation in special triangles like these streamlines the process. Understanding these underlying principles eliminates guesswork and enables precise, fast computations in exercises involving 30°-60°-90° triangles.