Problem 7 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) find the exact values of the remaining parts. $$b=5 \sqrt{3}, \quad c=10 \sqrt{3}$$

Step-by-Step Solution

Answer

Side \(a = 15\); angles \(\alpha = 30^{\circ}\), \(\beta = 60^{\circ}\).

Step 1: Understand the Given Elements of Triangle

In this problem, triangle \(ABC\) is a right triangle with right angle \(\gamma = 90^{\circ}\). We are also given that side \(b = 5\sqrt{3}\) and hypotenuse \(c = 10\sqrt{3}\). Our task is to find the unknown side \(a\) and the angles \(\alpha\) and \(\beta\).

Step 2: Use the Pythagorean Theorem

Since we have a right triangle, the Pythagorean Theorem applies: \[ a^2 + b^2 = c^2 \]Substitute the known values of \(b\) and \(c\):\[ a^2 + (5\sqrt{3})^2 = (10\sqrt{3})^2 \]Calculate the squares:\[ a^2 + 75 = 300 \]

Step 3: Solve for Side a

Rearrange the equation from Step 2:\[ a^2 = 300 - 75 \]\[ a^2 = 225 \]Take the square root of both sides:\[ a = \sqrt{225} = 15 \]

Step 4: Find Angle \(\alpha\) Using Trigonometry

We can use the sine ratio to find \(\alpha\):\[ \sin(\alpha) = \frac{b}{c} = \frac{5\sqrt{3}}{10\sqrt{3}} = \frac{1}{2} \]Since \(\sin(\alpha) = \frac{1}{2}\), \(\alpha = 30^{\circ}\).

Step 5: Find Remaining Angle \(\beta\)

Since \(\alpha + \beta = 90^{\circ}\) in a right triangle:\[ \beta = 90^{\circ} - 30^{\circ} = 60^{\circ} \]

Step 6: Verify Solution Consistency

Check the consistency of the answers:- Confirm \(a^2 + b^2 = c^2\) holds true: \[ 15^2 + (5\sqrt{3})^2 = 225 + 75 = 300 \], which equals \(300\).- Confirm angles: \(\alpha + \beta = 90^{\circ}\) with \(\gamma = 90^{\circ}\). Both conditions are satisfied.

Key Concepts

Pythagorean Theorem

In any right triangle, the Pythagorean Theorem is a fundamental principle. It relates the lengths of the sides of the triangle. According to this theorem, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.
The formula is given by:

• \[ a^2 + b^2 = c^2 \]

Let's delve deeper into understanding this with our given triangle. We know the hypotenuse, \(c\), and one side \(b\):

• \( b = 5\sqrt{3} \)
• \( c = 10\sqrt{3} \)

By substituting these values, we can solve for the missing side \(a\). We start by squaring the given side lengths, substitute into the formula, and then solve for \(a\). This process helps you analyze and understand the relationships between the triangle's sides.

Sine Function

The sine function is an essential trigonometric function used to relate the angle of a right triangle to the ratio of the length of the opposite side over the hypotenuse.
For an angle \(\alpha\) in our triangle:

• \[ \sin(\alpha) = \frac{\text{opposite side}}{\text{hypotenuse}} \]

In our specific case, we use the sine function to find \(\alpha\), knowing \(b\) is the opposite side when \(\alpha\) is the angle being evaluated. We apply:

• \[ \sin(\alpha) = \frac{5\sqrt{3}}{10\sqrt{3}} \]
• \[ \sin(\alpha) = \frac{1}{2} \]

Hence, the sine function shows us that \(\alpha = 30^{\circ}\) because \(\sin^{-1}(\frac{1}{2}) = 30^{\circ}\). Understanding the sine function is crucial because it gives us a method to find unknown angles and dimensions within right-angled triangles.

Angle Calculation

Determining the angles within a right triangle is made efficient through basic trigonometric identities. A right triangle has a total angle sum of \(180^{\circ}\). With one angle defined as \(90^{\circ}\), the sum of the other two must be \(90^{\circ}\).
Starting with finding angle \(\alpha\) using the sine function, we determined:

• \( \alpha = 30^{\circ} \)

With this information, the other angle \(\beta\) is computed simply by complementing \(\alpha\):

• \( \beta = 90^{\circ} - \alpha \)

Thus, for our example:

• \[ \beta = 90^{\circ} - 30^{\circ} = 60^{\circ} \]

Using this method allows for straightforward calculation and ensures that all angles within the right triangle are correctly identified.

Triangle Properties

Understanding triangle properties is vital when dealing with any geometric problem. Key properties of right triangles include:

• One angle is always \(90^{\circ}\).
• The sum of the angles is always \(180^{\circ}\).
• The hypotenuse is the longest side.

In our given problem, these properties prove essential:

• The triangle's right angle confirms that we can use the Pythagorean Theorem and trigonometric ratios effectively.
• We confirmed the sum of \(\alpha\) and \(\beta\) equals \(90^{\circ}\).

These fundamental properties help ensure problem-solving steps are logical and consistent. They provide a basis for calculating unknown sides and angles, ensuring that the geometric configurations and relationships are correctly maintained.