Problem 26 - Algebra and Trigonometry

Question

Solve each triangle. $$ a=4, \quad b=5, \quad c=3 $$

Step-by-Step Solution

Answer

Angles are approximately: \( \alpha = 53.13 ^\circ \), \( \beta = 90 ^\circ \), \( \gamma = 36.87 ^\circ \).

Step 1 - Verify Triangle Inequality

First, ensure the given sides form a valid triangle. The triangle inequality theorem states that the sum of any two sides must be greater than the third side. Check the following inequalities: 1. 4 + 5 > 32.4 + 3 > 53.5 + 3 > 4All inequalities are satisfied, so the side lengths form a valid triangle.

Step 2 - Use the Law of Cosines

The Law of Cosines helps find the angles of the triangle when all sides are known. It is given by: \[ c^2 = a^2 + b^2 - 2ab \cos \gamma \] Rearrange to solve for \( \gamma \): \[ \cos \gamma = \frac{a^2 + b^2 - c^2}{2ab} \] Substitute the values: \[ \cos \gamma = \frac{4^2 + 5^2 - 3^2}{2 \cdot 4 \cdot 5} = \frac{16 + 25 - 9}{40} = \frac{32}{40} = \frac{4}{5} \] So, \( \gamma = \cos^{-1} \left( \frac{4}{5} \right) \)

Step 3 - Calculate Angle \( \gamma \)

Use a calculator to find the inverse cosine: \[ \gamma = \cos^{-1} \left( \frac{4}{5} \right) \approx 36.87 ^\circ\]

Step 4 - Use the Law of Sines

With one angle known, use the Law of Sines to find another angle. The formula is: \[ \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \] First, solve for \( \alpha \): \[ \frac{4}{\sin(\alpha)} = \frac{3}{\sin(36.87 ^\circ)} \] Solve for \( \alpha \):\[ \sin(\alpha) = \frac{4 \cdot \sin(36.87 ^\circ)}{3} = \frac{4 \cdot 0.6}{3} = 0.8 \] So, \( \alpha = \sin^{-1}(0.8) \approx 53.13 ^\circ \)

Step 5 - Find Angle \( \beta \)

The sum of angles in a triangle is always \( 180 ^\circ \). So, solve for \( \beta \): \[ \beta = 180 ^\circ - \alpha - \gamma = 180 ^\circ - 53.13 ^\circ - 36.87 ^\circ = 90 ^\circ \]

Key Concepts

Triangle Inequality Theorem

A fundamental concept in triangle solving is ensuring that the given sides can indeed form a triangle. This is guaranteed by the Triangle Inequality Theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
For example:

• We have sides: 4, 5, and 3.


• Check if 4 + 5 > 3 (Yes, because 9 > 3).


• Next, 4 + 3 > 5 (Yes, because 7 > 5).


• Finally, 5 + 3 > 4 (Yes, because 8 > 4).

Since all these conditions are satisfied, the given sides form a valid triangle.

Law of Cosines

The Law of Cosines is incredibly useful when you know all three sides of a triangle and need to find its angles. It's an extension of the Pythagorean theorem, applicable to any triangle, not just right ones.
The formula is:

Law of Sines

Once you know at least one angle and its opposite side, paired with another side, you can use the Law of Sines to find missing angles or sides. The formula looks like this:
The Law of Sines states that the sides of a triangle are proportional to the sines of their opposite angles.
From the exercise, we used it to find another angle once one was known:

• {a}/{sin(α)} = {b}/{sin(β)} = {c}/{sin(γ)}


• To find a missing angle, rearrange and solve. For example, sin(α) = 4 sin(α) = (4 * sin(36.87°)) / 3, leading to sin(α) 1) Calculate sin(α) 2) inverse sin to find α

This is a great tool for cases when data is limited to one side’s pairing with its angle.

Angles in a Triangle

The sum of the internal angles in any triangle is always 180°. This is a basic but vital principle to solve or verify the angles within any given triangle.
For instance, once you find two angles, the third can easily be calculated using this rule: {α + β + γ = 180°}
In the exercise solution, it was used as:
β = 180° − α − γ
If α is 53.13° and γ is 36.87°, then β equals:
β = 180° − 53.13° − 36.87° = 90°
This helps to cross-check all calculations and ensure your triangle is accurately solved.