Problem 4 - College Trigonometry

Question

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ a=3, b=4, \gamma=90^{\circ} $$

Step-by-Step Solution

Answer

The remaining side \(c\) is 5, and angles \(\alpha \approx 36.87^{\circ}\) and \(\beta \approx 53.13^{\circ}\).

Step 1: Understanding the Law of Cosines

The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\) opposite to angles \(\alpha\), \(\beta\), and \(\gamma\) respectively, \(c^2 = a^2 + b^2 - 2ab \cos(\gamma)\). Given that \(\gamma = 90^{\circ}\), this reduces to the Pythagorean theorem because \(\cos(90^{\circ}) = 0\). Thus, the equation simplifies to \(c^2 = a^2 + b^2\).

Step 2: Substituting Known Values

We have \(a = 3\), \(b = 4\), and \(\gamma = 90^{\circ}\). Substitute these values into the simplified Law of Cosines equation: \[c^2 = 3^2 + 4^2\].

Step 3: Calculating \(c^2\)

Calculate \(c^2\) using the substituted values: \[c^2 = 9 + 16 = 25\].

Step 4: Finding the Length of Side \(c\)

To find \(c\), take the square root of \(25\), giving \(c = 5\).

Step 5: Finding the Other Angles

Since the triangle is a right triangle with \(\gamma = 90^{\circ}\), we use the definitions of trigonometric ratios like \(\sin\), \(\cos\), and \(\tan\) to find the other angles. Use \(\tan(\alpha) = \frac{3}{4}\) and \(\tan(\beta) = \frac{4}{3}\) to find \(\alpha\) and \(\beta\) respectively.

Step 6: Calculating Angles \(\alpha\) and \(\beta\)

Using \(\tan(\alpha) = \frac{3}{4}\), find \(\alpha\) using the inverse tangent function: \(\alpha \approx \tan^{-1}\left(\frac{3}{4}\right)\). This gives \(\alpha \approx 36.87^{\circ}\). Similarly, \(\beta \approx 53.13^{\circ}\) using \(\tan(\beta) = \frac{4}{3}\).

Key Concepts

Right Triangle

A right triangle is a special type of triangle where one of the angles is exactly 90 degrees. This defining characteristic creates interesting relationships between the triangle's sides and angles. When solving a right triangle, we can leverage these unique properties to find unknown sides or angles efficiently.
In our problem, we have a right triangle with sides labeled as \(a = 3\), \(b = 4\), and a right angle \(\gamma = 90^{\circ}\). Given these inputs, the task is to determine the length of the hypotenuse, typically represented as side \(c\), and any other unknown angles. Right triangles frequently lead us to use specific mathematical principles such as trigonometric ratios and the Pythagorean theorem, which gives us the tools we need to find solutions efficiently.

Trigonometric Ratios

Trigonometric ratios are functions that relate the angles of a triangle to the lengths of its sides. In a right triangle, the primary trigonometric ratios are sine, cosine, and tangent:

• Sine (\(\sin\)): This is the ratio of the opposite side to the hypotenuse.
• Cosine (\(\cos\)): This is the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\)): This is the ratio of the opposite side to the adjacent side.

For the angle \(\alpha\), we use \(\tan(\alpha) = \frac{3}{4}\) because it describes the relationship between the opposite side (3) and the adjacent side (4). Similarly, for angle \(\beta\), \(\tan(\beta) = \frac{4}{3}\). These relationships allow us to calculate the angles with ease using inverse trigonometric functions, which are commonly available on scientific calculators.

Pythagorean Theorem

The Pythagorean theorem is a fundamental theorem in geometry, used exclusively in right triangles. It states that the square of the length of the hypotenuse (side \(c\)) is equal to the sum of the squares of the other two sides \(a\) and \(b\). Mathematically, this is expressed as:
\[ c^2 = a^2 + b^2 \]
In our example, we substitute \(a = 3\) and \(b = 4\) into this formula to find \(c\): \[ c^2 = 3^2 + 4^2 = 9 + 16 = 25 \]
Solving for \(c\) involves taking the square root of 25, yielding \(c = 5\).
This theorem, as simple as it is, remains a powerful tool for solving problems related to right triangles, and understanding it can greatly enhance your ability to work with triangular shapes.

Angle Calculation

Calculating angles in a right triangle typically involves using trigonometric functions and their inverses. Once the sides of the triangle are known, determining the angles can illuminate the complete structure of the triangle. In our problem:
- We use \(\tan(\alpha) = \frac{3}{4}\) to find \(\alpha\). Inputting this into an inverse tangent function (\(\tan^{-1}\)) gives \(\alpha \approx 36.87^{\circ}\).- For \(\beta\), we have \(\tan(\beta) = \frac{4}{3}\), which yields \(\beta \approx 53.13^{\circ}\) when calculated.
Combining these calculations with the known 90-degree angle, this ensures that the sum of angles within a triangle always equals 180 degrees. Accurate angle calculation helps confirm the dimensions of the triangle and plays a crucial role in real-world applications.