Problem 10 - 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. $$ \alpha=30^{\circ}, a=7, b=14 $$

Step-by-Step Solution

Answer

\( \beta = 90^{\circ}, \gamma = 60^{\circ}, c = 7\sqrt{3} \).

Step 1: Identify Known Values

We know that \( \alpha = 30^{\circ}, a = 7 \), and \( b = 14 \). We need to find \( \beta, \gamma, \text{and} \; c \).

Step 2: Use the Law of Sines

The Law of Sines states: \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)} \). Therefore, \( \frac{7}{\sin(30^{\circ})} = \frac{14}{\sin(\beta)} \).

Step 3: Solve for \( \beta \)

We know \( \sin(30^{\circ}) = \frac{1}{2} \). Substitute into the equation: \( \frac{7}{\frac{1}{2}} = \frac{14}{\sin(\beta)} \). Simplifying, we have \( 14 = \frac{14}{\sin(\beta)} \). Thus, \( \sin(\beta) = 1 \), giving \( \beta = 90^{\circ} \).

Step 4: Find \( \gamma \)

The sum of angles in a triangle is \( 180^{\circ} \). Therefore, \( \gamma = 180^{\circ} - 30^{\circ} - 90^{\circ} = 60^{\circ} \).

Step 5: Calculate Side \( c \) Using the Law of Sines

Using \( \frac{a}{\sin(\alpha)} = \frac{c}{\sin(\gamma)} \), we substitute \( a = 7, \alpha = 30^{\circ}, \text{and} \; \gamma = 60^{\circ} \). \( \frac{7}{1/2} = \frac{c}{\sqrt{3}/2} \). Solving, \( c = 7\sqrt{3} \).

Key Concepts

Angles in a Triangle

Understanding angles in a triangle is essential for solving any trigonometric problem related to triangles. A triangle is a three-sided polygon, and the sum of all interior angles in any triangle always equals 180 degrees. This rule is vital because it helps in finding the unknown angles when given some of the angles.
In our given problem, we know that one angle is 30 degrees and another angle is 90 degrees, because we deduced it using the Law of Sines. Once we have two angles, finding the third angle is straightforward. Since all angles in a triangle sum up to 180 degrees, the third angle can be found using simple subtraction: 180 minus the known angles.
This principle is particularly useful in many mathematical problems and helps to ensure that the triangle properties remain consistent. Remembering this rule allows you to find unknown angles quickly and easily.

Solving Triangles

When we talk about solving triangles, we refer to finding the unknown sides and angles of the triangle. This involves different strategies or formulas like the Law of Sines or the Law of Cosines, depending on the information provided. Solving triangles can be approached systematically by:

• Identifying the given and required values
• Applying trigonometric laws and properties
• Solving equations to find unknown values

In our exercise, we started with given values for one side and two angles, then applied the Law of Sines to find the missing parts.
The process can often lead to determining whether the triangle is right-angled, isosceles, or scalene based on the side lengths and angles. Solving triangles necessitates an understanding of various trigonometric principles and a methodical approach to solution finding.

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles and sides of a triangle. The primary trigonometric functions are sine, cosine, and tangent, which are frequently used in solving triangles. Each function represents a ratio of sides in a right-angled triangle.
In our scenario, the Law of Sines uses the sine function:\[\frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}\] This equation allows us to relate the ratios of sides to their opposite angles, offering a powerful tool for finding unknown values in triangles.
Recognizing and using these trigonometric functions accurately is key in not just problems involving right triangles but any triangle-solving endeavor. Understanding how these functions work together also builds a solid foundation in both geometry and trigonometry.