Problem 33 - Intermediate Algebra

Question

Find the missing side lengths in each triangle. Give the exact answer and then an approximation to two decimal places, when appropriate. See Example 4 In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the length of the leg opposite the \(30^{\circ}\) angle is \(75 \mathrm{cm} .\) Find the length of the leg opposite the \(60^{\circ}\) angle and the length of the hypotenuse. Give the exact answer and then an approximation to two decimal places, when appropriate.

Step-by-Step Solution

Answer

The side opposite \(60^{\circ}\) is exactly \(75\sqrt{3}\, \mathrm{cm}\), approximately \(129.90\, \mathrm{cm}\), and the hypotenuse is \(150\, \mathrm{cm}\).

Step 1: Identify Triangle Properties

In a \(30^{\circ}-60^{\circ}-90^{\circ}\) triangle, the sides are in the ratio \(1 : \sqrt{3} : 2\). The side opposite \(30^{\circ}\) is the smallest, the side opposite \(60^{\circ}\) is \(\sqrt{3}\) times the smallest side, and the hypotenuse is 2 times the smallest side.

Step 2: Determine the Length of the Hypotenuse

The length of the side opposite the \(30^{\circ}\) angle is given as \(75\, \mathrm{cm}\). According to the triangle's properties, the hypotenuse should be twice this length:\[\text{Hypotenuse} = 2 \times 75\, \mathrm{cm} = 150\, \mathrm{cm} \]

Step 3: Calculate the Opposite Side of the 60° Angle

The leg opposite the \(60^{\circ}\) angle is \(\sqrt{3}\) times the leg opposite the \(30^{\circ}\) angle:\[\text{Length of the side opposite } 60^{\circ} = 75\, \mathrm{cm} \times \sqrt{3} \approx 75 \times 1.732 \approx 129.9\, \mathrm{cm} \]

Step 4: Provide Exact and Approximate Answers

The exact length of the leg opposite the \(60^{\circ}\) angle is \(75 \sqrt{3}\, \mathrm{cm}\), and its approximate length to two decimal places is \(129.90\, \mathrm{cm}\). The hypotenuse is exactly \(150\, \mathrm{cm}\), and since that is already a whole number, it remains the same in its approximate form.

Key Concepts

Right Triangles

Right triangles are fundamental in trigonometry, characterized by having one angle exactly equal to 90 degrees. This makes them particularly interesting in calculations and problem-solving. A right triangle consists of two legs and a hypotenuse, which is the side opposite the right angle and also the longest side of the triangle. Understanding right triangles helps us solve various practical problems involving trigonometric functions and ratios.
A unique aspect of right triangles is their ability to relate different mathematical concepts like angles, sides, and functions seamlessly.
In any right triangle, the Pythagorean theorem, trigonometric ratios, and angle properties come into play, allowing learners to connect geometry with algebraic functions.
Right triangles are present in many real-life situations, like ramps, stairs, and even in architectural designs, making them essential for students to understand.

Trigonometric Ratios

Trigonometric ratios are crucial for studying relationships between side lengths and angles in right triangles. These ratios help us calculate unknown angles or sides of the triangle efficiently.
The three primary trigonometric ratios are:

• Sine ( sin) - the ratio of the opposite side to the hypotenuse.
• Cosine ( cos) - the ratio of the adjacent side to the hypotenuse.
• Tangent ( tan) - the ratio of the opposite side to the adjacent side.

Each ratio provides a way to link an angle of a right triangle to specific side lengths.
Moreover, special triangles like the 30-60-90 degree triangles have specific ratios. In these triangles, for instance, you know in advance the sides are in the ratio 1: sqrt{3}:2, making calculations predictable and straightforward.
Understanding these ratios is key for solving not only textbook problems but also real-world scenarios where indirect measurement is necessary.

Angle Properties

In right triangles, angle properties help define the relationships between the triangle's angles and sides. A fundamental property is that the sum of all angles in any triangle, including right triangles, is always 180 degrees. With one angle being 90 degrees, the other two angles must sum up to 90 degrees as well.
Specific right triangles, such as the 30-60-90 and 45-45-90 triangles, have fixed angle measures with predictable side length relationships.
In a 30-60-90 triangle, for instance:

• The angle opposite the shortest side is 30 degrees.
• The angle opposite the medium side is 60 degrees.
• The angle opposite the hypotenuse is naturally 90 degrees.

These angle properties and their consistent external/internal relationships are invaluable when calculating missing side lengths or angles, especially when using trigonometric ratios and formulas.

Pythagorean Theorem

The Pythagorean Theorem is a staple of geometry, especially when it comes to right triangles. The theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides. This can be mathematically expressed as:\[a^2 + b^2 = c^2\] where \(a\) and \(b\) are the legs of the triangle, and \(c\) is the hypotenuse.
This theorem is a powerful tool for determining unknown sides in right triangles. For example, you can easily find the hypotenuse if the lengths of the other two sides are known, or vice versa.
It is important to note how the Pythagorean Theorem interplays with trigonometric ratios and angle properties, providing a comprehensive framework for solving right triangle problems. The theorem is not only fundamental in theoretical mathematics but also in many fields such as engineering and computer science, where precise measurements and calculations are required.