Problem 16 - Precalculus: Functions and Graphs

Question

Given the indicated parts of triangle \(A B C\) with \(\gamma=90^{\circ},\) approximate the remaining parts. $$a=0.42, \quad c=0.68$$

Step-by-Step Solution

Answer

Side \(b \approx 0.535\), \(\angle A \approx 38.08^{\circ}\), and \(\angle B \approx 51.92^{\circ}\).

Step 1: Identify the Type of Triangle

Since triangle \(ABC\) has \(\gamma=90^{\circ}\), it is a right triangle with \(\angle C\) as the right angle. The given sides are \(a\) and \(c\) where \(c\) is the hypotenuse.

Step 2: Calculate Side b Using Pythagorean Theorem

For a right triangle, the Pythagorean theorem states that \(a^2 + b^2 = c^2\). Plugging in the given values, we have: \[ (0.42)^2 + b^2 = (0.68)^2 \] Solve for \(b^2\):\[ b^2 = 0.68^2 - 0.42^2 \]Calculate \(0.68^2 = 0.4624\) and \(0.42^2 = 0.1764\).Then, \(b^2 = 0.4624 - 0.1764 = 0.286\). Taking the square root gives \(b \approx \sqrt{0.286} \approx 0.535\).

Step 3: Calculate Remaining Angles Using Trigonometry

\(\angle A\) can be found using sine. \( \sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{0.42}{0.68} \). Thus, \( \alpha = \sin^{-1}(0.6176) \approx 38.08^{\circ}\).Since \(\gamma=90^{\circ}\) and the sum of angles in a triangle is \(180^{\circ}\), \(\beta = 90^{\circ} - 38.08^{\circ} \approx 51.92^{\circ}\).

Key Concepts

Right Triangle

In geometry, understanding the properties of a right triangle is essential. A right triangle is distinguished by one angle being exactly 90 degrees, known as the right angle. This angle forms a perfect "L" shape, making it easy to identify visually. In the case of triangle \(ABC\), we see that \(\gamma = 90^{\circ}\), making \(\angle C\) the right angle. Right triangles have two legs and a hypotenuse, which is the side opposite the right angle. In our example, side \(c = 0.68\) is the hypotenuse, while side \(a = 0.42\) is one of the legs. Knowing the hypotenuse and one leg allows us to explore several calculations and properties related to right triangles.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics that applies exclusively to right triangles. It provides a relationship between the lengths of the sides in a right triangle. The theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Mathematically, this is expressed as: \(a^2 + b^2 = c^2\). Here, given \(a = 0.42\) and \(c = 0.68\), we can calculate the missing side \(b\) by rearranging the theorem to solve for \(b^2\):

• First, calculate \(0.68^2 = 0.4624\).
• Then, calculate \(0.42^2 = 0.1764\).
• Next, subtract to find \(b^2\): \(b^2 = 0.4624 - 0.1764 = 0.286\).
• Finally, take the square root: \(b \approx \sqrt{0.286} \approx 0.535\).

This theorem is powerful because once you know any two sides of a right triangle, you can always find the third.

Angle Calculation

Calculating angles in a right triangle involves using trigonometric functions and the properties of triangles. In a triangle, the sum of the internal angles adds up to \(180^{\circ}\). With \(\gamma = 90^{\circ}\) in triangle \(ABC\), we need to find \(\alpha\) and \(\beta\). Using the sine function, which relates angles to side lengths, we can find \(\alpha\). The formula \(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\) is utilized. Here, \(\alpha\) is opposite side \(a\):

• \(\sin(\alpha) = \frac{0.42}{0.68}\)
• Calculate this to be approximately \(0.6176\).
• Thus, \(\alpha \approx \sin^{-1}(0.6176) \approx 38.08^{\circ}\).

With \(\alpha\) known, \(\beta\) can be found as \(\beta = 90^{\circ} - \alpha\), giving us \(\beta \approx 51.92^{\circ}\).

Sine Function

The sine function is one of the primary trigonometric functions, commonly used in angle calculation within triangles. For a given angle in a right triangle, the sine function relates this angle to the ratio of the side of the triangle opposite the angle to the hypotenuse. This is expressed as \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\). In our context, calculating \(\alpha\) in triangle \(ABC\) uses the sine function.

• The opposite side to \(\alpha\) is \(a = 0.42\), and the hypotenuse is \(c = 0.68\).
• Therefore the sine of \(\alpha\) is \(\sin(\alpha) = \frac{0.42}{0.68} = 0.6176\).
• To find the angle, take the inverse sine: \(\alpha = \sin^{-1}(0.6176) \approx 38.08^{\circ}\).

This highlights how the sine function can be utilized to deduce angles in right triangles from side lengths, offering a powerful tool in trigonometry.