Problem 13 - Algebra and Trigonometry

Question

13-18 $$ Sketch each triangle, and then solve the triangle using the Law of Sines. $$ \angle A=50^{\circ}, \quad \angle B=68^{\circ}, \quad c=230 $$

Step-by-Step Solution

Answer

\( a \approx 201.13 \), \( b \approx 241.32 \), \( C = 62^{\circ} \).

Step 1: Sketch the Triangle

Draw a triangle labeling vertices as \( A \), \( B \), and \( C \). Mark angle \( A \) as \( 50^{\circ} \), angle \( B \) as \( 68^{\circ} \), and side \( c \) opposite angle \( C \) as 230. The angle \( C \), opposite side \( c \), is yet to be determined.

Step 2: Find Angle C

Use the angle sum property of a triangle: \[ C = 180^{\circ} - A - B = 180^{\circ} - 50^{\circ} - 68^{\circ} = 62^{\circ} \].

Step 3: Apply the Law of Sines

The Law of Sines states \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Since \( c = 230 \), \( \sin C = \sin(62^{\circ}) \), solve for sides \( a \) and \( b \).For \( a \), use: \[ \frac{a}{\sin(50^{\circ})} = \frac{230}{\sin(62^{\circ})} \Rightarrow a = \frac{230 \cdot \sin(50^{\circ})}{\sin(62^{\circ})} \].For \( b \), use: \[ \frac{b}{\sin(68^{\circ})} = \frac{230}{\sin(62^{\circ})} \Rightarrow b = \frac{230 \cdot \sin(68^{\circ})}{\sin(62^{\circ})} \].

Step 4: Calculate Side Lengths

Calculate the values:\( a = \frac{230 \cdot \sin(50^{\circ})}{\sin(62^{\circ})} \approx 201.13 \) (rounded to two decimal places)\( b = \frac{230 \cdot \sin(68^{\circ})}{\sin(62^{\circ})} \approx 241.32 \) (rounded to two decimal places)

Key Concepts

Triangle Properties

Triangles are fundamental geometric shapes with three sides and three angles, adding up to make a crucial foundation in geometry. Each triangle has unique characteristics that can help determine unknown values, such as side lengths and angles. Understanding these properties simplifies the process of solving triangles.

• Sides and Angles: Triangles have three sides and three angles, and the notation typically uses capital letters for angles and lowercase letters for sides opposite those angles (e.g., angle A is opposite side a).
• Types of Triangles: Depending on their angles, triangles could be classified as acute, right, or obtuse. In terms of sides, they can be scalene, isosceles, or equilateral.
• Congruence and Similarity: Triangles can be congruent (exactly the same size and shape) or similar (same shape but different sizes), and these properties help solve many geometric problems.

When solving a triangle, knowing the type and properties can guide which rules or laws, like the Law of Sines or Cosines, to apply. This is key in determining unknown sides or angles when only limited information is provided.

Angle Sum Property

One of the fundamental properties of triangles is the angle sum property. This principle states that the sum of all interior angles in a triangle is always 180 degrees. This rule applies to every triangle, no matter its type or size.

In practice, if you know two angles of a triangle, you can always find the third angle using this property:

• Use the formula: \[ C = 180^{\circ} - A - B \]
• For example, in our exercise, angle A is 50 degrees, and angle B is 68 degrees. By subtracting these values from 180 degrees, we find angle C:\[ C = 180^{\circ} - 50^{\circ} - 68^{\circ} = 62^{\circ} \]
• This completes the determination of all angles in the triangle, which is a crucial step in further computations such as applying the Law of Sines or solving for side lengths.

This property is not only a basic part of geometry but also a crucial step in solving more complex problems involving triangles.

Solving Triangles

Solving triangles involves finding unknown side lengths and angle measures using given information through geometric principles like the Law of Sines or the Law of Cosines.

In our exercise, we use the Law of Sines, which relates the sides of a triangle to the sines of its angles:

• The formula is: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]
• By using given side and angle values, you can plug them into the equation to find missing side lengths. Because side c and angle C are known, and angles A and B have been calculated, you can determine sides a and b.
• For side a, the calculation is: \[ a = \frac{230 \cdot \sin(50^{\circ})}{\sin(62^{\circ})} \approx 201.13 \]
• For side b, the calculation is: \[ b = \frac{230 \cdot \sin(68^{\circ})}{\sin(62^{\circ})} \approx 241.32 \]

By applying these relations, you can solve for unknown values efficiently. The Law of Sines is especially useful for non-right triangles where direct measurements aren't available. It provides a straightforward method to find unknowns, maintaining the connection between the triangle's angular and linear dimensions.