Problem 20 - College Trigonometry

Question

Solve for the remaining side(s) and angle(s) if possible. As in the text, \((\alpha, a)\), \((\beta, b)\) and \((\gamma, c)\) are angle-side opposite pairs. $$ \alpha=50^{\circ}, a=25, b=12.5 $$

Step-by-Step Solution

Answer

\(\beta \approx 22.5^{\circ}, \gamma \approx 107.5^{\circ}, c \approx 31.3\).

Step 1: Identify Known and Unknowns

You are given \( \alpha = 50^{\circ}, a = 25, b = 12.5 \). You need to find the unknown angle \( \beta \) and sides \( c \) and angle \( \gamma \).

Step 2: Use Sine Rule for Angle \(\beta\)

Apply the Sine Rule: \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \). Substitute the known values: \( \frac{25}{\sin 50^{\circ}} = \frac{12.5}{\sin \beta} \). Calculate \( \sin \beta = \frac{12.5 \times \sin 50^{\circ}}{25} \). Compute \( \sin \beta \approx 0.383 \).

Step 3: Determine Angle \(\beta\)

Find angle \( \beta \) by calculating its inverse sine: \( \beta = \sin^{-1}(0.383) \). Compute \( \beta \approx 22.5^{\circ} \).

Step 4: Determine Angle \(\gamma\)

Use the angle sum property of triangles: \( \alpha + \beta + \gamma = 180^{\circ} \). Substitute the known angles to find \( \gamma \): \( 50^{\circ} + 22.5^{\circ} + \gamma = 180^{\circ} \). Solve for \( \gamma \): \( \gamma = 180^{\circ} - 72.5^{\circ} = 107.5^{\circ} \).

Step 5: Use Sine Rule for Side \(c\)

Apply the Sine Rule again: \( \frac{a}{\sin \alpha} = \frac{c}{\sin \gamma} \). Substitute known values: \( \frac{25}{\sin 50^{\circ}} = \frac{c}{\sin 107.5^{\circ}} \). Calculate \( c = \frac{25 \times \sin 107.5^{\circ}}{\sin 50^{\circ}} \). Compute \( c \approx 31.3 \).

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that helps us find unknown angles and sides of a triangle. It states that the ratio of a side to the sine of its opposite angle is the same for all three sides of a triangle. This can be expressed as:

• \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \)

This makes it incredibly useful when you know:

• Two angles and one side (AAS or ASA scenario), or
• Two sides and one non-included angle (SSA scenario).

In our original exercise, we used this law to find the unknown angle \( \beta \) by setting up the equation \( \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} \). By substituting the known values, we found \( \sin \beta \) and then applied inverse trigonometric functions to find \( \beta \). This systematic use of the law of sines is essential in solving many triangle-related problems.

Triangle Properties

Triangles have a few vital properties that are crucial for solving many problems in trigonometry:

• The sum of all interior angles in a triangle is always \( 180^{\circ} \).
• In any triangle, the side opposite the larger angle is longer.
• The Triangle Inequality Theorem states that the sum of the lengths of any two sides must be greater than the length of the third side.

Understanding these properties helps when verifying the plausibility of calculated angles and side lengths.
In our task, the sum of angles property was used to find the third angle \( \gamma \), ensuring that the triangle conforms to basic angle rules. All these properties are fundamental cornerstones in geometry and are often the backbone of more complex trigonometric solutions.

Angle Sum Property

The angle sum property is a simple yet powerful concept in trigonometry. It states that the total of the internal angles in any triangle equals \( 180^{\circ} \).
This rule can be efficiently used to find an unknown angle if you already know the other two.
For example, in the exercise provided, after we found the angle \( \beta \), we utilized this property to determine \( \gamma \) by subtracting the sum of \( \alpha \) and \( \beta \) from \( 180^{\circ} \):

• \( \alpha + \beta + \gamma = 180^{\circ} \)
• \( \gamma = 180^{\circ} - (\alpha + \beta) \)

Such calculations are straightforward but highlight the importance of basic geometric principles. They're a reliable tool for verifying calculations and ensuring accuracy in solving triangle puzzles.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential tools for finding angles when you know the value of their sine, cosine, or tangent.
In trigonometry, these functions allow us to determine angles from ratios.

• \( \sin^{-1}(x) \)
• \( \cos^{-1}(x) \)
• \( \tan^{-1}(x) \)

These notations represent the inverse operations to the trigonometric functions themselves. In our scenario, after finding \( \sin \beta \), the inverse sine function \( \sin^{-1} \) was used to calculate the angle \( \beta \).
This is a typical application when solving for angles in a triangle, providing the bridge between the numerical ratios and the angle measures themselves. Understanding and using these inverse functions are key skills for mastering trigonometric applications in geometry.