Problem 37 - Intermediate Algebra

Question

Find the missing lengths in each triangle. Give the exact answer and then an approximation to two decimal places. See Example 5. In a \(30^{\circ}-60^{\circ}-90^{\circ}\) right triangle, the length of the leg opposite the \(60^{\circ}\) angle is 55 millimeters. Find the length of the leg opposite the \(30^{\circ}\) angle and the length of the hypotenuse. Give the exact answer and then an approximation to two decimal places.

Step-by-Step Solution

Answer

Leg opposite 30°: exact \(\frac{55\sqrt{3}}{3}\), approx. 31.82 mm; Hypotenuse: exact \(\frac{110\sqrt{3}}{3}\), approx. 63.64 mm.

Step 1: Understand Triangular Ratios

In a 30-60-90 triangle, the ratios of the sides are 1 (opposite the 30° angle), \(\sqrt{3}\) (opposite the 60° angle), and 2 (the hypotenuse). These ratios can be used to find unknown side lengths.

Step 2: Use the Ratio to Find the Opposite to 30°

Given that the leg opposite the 60° angle is 55 mm, use the ratio \(\sqrt{3}:1\) to find the length of the leg opposite the 30° angle. Set up the equation: \(\frac{x}{\sqrt{3}} = \frac{55}{1}\). Solve for \(x\), which gives \(x = \frac{55}{\sqrt{3}}\).

Step 3: Simplify to Find Exact Opposite to 30°

Rationalize the denominator to find the exact length: \(x = \frac{55}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{55\sqrt{3}}{3}\). This is the exact length of the leg opposite the 30° angle.

Step 4: Calculate the Hypotenuse

Using the 30°-60°-90° triangle ratio, the hypotenuse is twice the length of the leg opposite the 30° angle: \(h = 2 \times \frac{55\sqrt{3}}{3} = \frac{110\sqrt{3}}{3}\). This gives the exact length of the hypotenuse.

Step 5: Approximate the Leg Opposite 30°

To find the approximate length, calculate \(\frac{55\sqrt{3}}{3} \approx 31.82\) millimeters by using a calculator to evaluate \(\sqrt{3} \approx 1.732\).

Step 6: Approximate the Hypotenuse

To find the approximate length of the hypotenuse, calculate \(\frac{110\sqrt{3}}{3} \approx 63.64\) millimeters, again using the approximation \(\sqrt{3} \approx 1.732\).

Key Concepts

Right Triangle Ratios

In every right triangle, there are specific relationships between the angles and the side lengths. One of the most common special right triangles is the 30-60-90 triangle. The side lengths of a 30-60-90 triangle are in the ratio 1 : \( \sqrt{3} \) : 2. This means:

• The shortest side, opposite the 30° angle, is the baseline unit of comparison, represented as 1.
• The length opposite the 60° angle is \( \sqrt{3} \) times the shortest side.
• The hypotenuse is exactly 2 times the shortest side.

By understanding these ratios, you can easily find any side length if you know just one of them. For instance, if you know the side opposite the 60° angle, you can find the others by maintaining the given ratio. These ratios reflect the consistent relationships in a 30-60-90 triangle.

Trigonometry

Trigonometry involves the relationships between angles and sides of triangles. In a 30-60-90 triangle, the constants \( \sqrt{3} \) and 2 are derived from the specific properties of the angles. These constants allow us to use trigonometric ratios effectively to solve problems.
Using trigonometric ratios such as sine, cosine, and tangent can be very helpful in different scenarios:

• Sine (sin) of an angle is the opposite side over the hypotenuse.
• Cosine (cos) is the adjacent side over the hypotenuse.
• Tangent (tan) is the opposite side over the adjacent side.

In a 30-60-90 triangle, these trigonometric functions can be simplified using the known ratios:

• \( \text{sin}(30°) = \frac{1}{2} \)
• \( \text{cos}(30°) = \frac{\sqrt{3}}{2} \)
• \( \text{tan}(30°) = \frac{1}{\sqrt{3}} \)

Understanding these ratios and how they relate to angles in triangles is key to applying trigonometry effectively.

Rationalizing the Denominator

Rationalizing the denominator is an important skill in mathematical simplification. It transforms expressions into a more standard form without radicals in the denominator. This concept is especially useful in simplifying answers involving radicals, such as the side lengths in right triangles like the 30-60-90.
When you have a fraction where the denominator contains a radical, multiplying the numerator and denominator by the radical can help. This process gets rid of the radical in the denominator:

• For example, if you have \( \frac{55}{\sqrt{3}} \), multiply both the numerator and the denominator by \( \sqrt{3} \) to get \( \frac{55\sqrt{3}}{3} \).

This form is more simplified and easier to interpret when using in further calculations. By rationalizing the denominator, you make sure your expressions are simpler and more universally accepted in mathematics.