Problem 14 - Algebra and Trigonometry

Question

13-18 $$ Sketch each triangle, and then solve the triangle using the Law of Sines. $$ \angle A=23^{\circ}, \quad \angle B=110^{\circ}, \quad c=50 $$

Step-by-Step Solution

Answer

The triangle has sides approximately \( a = 26.57 \), \( b = 66.07 \), and \( c = 50 \).

Step 1: Sketch the Triangle

To begin, we sketch a triangle labeled as \( \triangle ABC \). Given, \( \angle A = 23^{\circ} \) and \( \angle B = 110^{\circ} \). Place \( \angle A \) at one vertex and \( \angle B \) at an adjacent vertex, placing \( \angle C \) at the remaining vertex. Thus, the third angle, \( \angle C \), is opposite side \( c = 50 \).

Step 2: Find the Missing Angle

Since we know the sum of angles in a triangle is \( 180^{\circ} \), calculate the missing angle \( \angle C \) using the equation: \( \angle C = 180^{\circ} - \angle A - \angle B \). Plug in the known angles: \( \angle C = 180^{\circ} - 23^{\circ} - 110^{\circ} = 47^{\circ} \).

Step 3: Apply the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We use this to find side \( a \) and side \( b \).

Step 4: Solve for Side a

Using \( \frac{a}{\sin A} = \frac{c}{\sin C} \), substitute \( c = 50 \), \( \angle A = 23^{\circ} \), and \( \angle C = 47^{\circ} \): \[ a = \frac{50 \cdot \sin 23^{\circ}}{\sin 47^{\circ}} \]. Calculate \( a \) using a calculator: \( a \approx 26.57 \).

Step 5: Solve for Side b

Using \( \frac{b}{\sin B} = \frac{c}{\sin C} \), substitute \( c = 50 \), \( \angle B = 110^{\circ} \), and \( \angle C = 47^{\circ} \): \[ b = \frac{50 \cdot \sin 110^{\circ}}{\sin 47^{\circ}} \]. Calculate \( b \) using a calculator: \( b \approx 66.07 \).

Key Concepts

Triangle Properties

Every triangle has a unique set of properties that can help us understand and solve for its unknown elements. A triangle consists of three sides, three angles, and certain relationships these parts share with each other. Triangles can be categorized based on their angles (such as acute, right, or obtuse) or by their side lengths (like equilateral, isosceles, or scalene).

Key properties include:

• The sum of the angles of any triangle is always 180°.
• The exterior angle of a triangle is equal to the sum of the two opposite interior angles.
• In any triangle, the longest side is opposite the largest angle, and the shortest side is opposite the smallest angle.

Understanding these properties can greatly assist us in solving for unknown sides and angles, especially when using rules such as the Law of Sines or Law of Cosines.

Angle Sum in a Triangle

The concept of angle sum is crucial when working with triangles. In a simple plane triangle, the sum of all interior angles is always 180 degrees.

Calculating the Missing Angle

To find a missing angle in a triangle when two angles are known, you apply the formula:

• Add the two known angles together.
• Subtract the sum from 180°.

For example, in triangle ABC where angles A and B are given as 23° and 110°, respectively, the missing angle C can be found by:\(\angle C = 180^{\circ} - (\angle A + \angle B) = 180^{\circ} - (23^{\circ} + 110^{\circ}) = 47^{\circ}\)

Finding this missing angle allows us to use further calculations to solve the rest of the triangle.

Solving Triangles

Solving a triangle involves determining all its unknown sides and angles. This often requires a combination of properties, rules, and equations.

Steps to Solve a Triangle using the Law of Sines

1. **Identify known values:** Normally, at least one side and two angles or two sides and one angle must be known.2. **Find missing angle:** Use the angle sum property to find the unknown angle if needed.3. **Use Law of Sines:** Use the formula \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] This helps find unknown side lengths when paired with their respective opposite angles.4. **Calculate unknown sides:** Rearrange the equation and solve for missing side lengths using a calculator for trigonometric functions like \( \sin \).

It is often beneficial to sketch the triangle and label known values, which aids in visualizing relationships and ensuring no errors in calculation.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are essential in solving triangles. Each function relates the angles of a triangle to its sides, making them pivotal tools in trigonometry.The Law of Sines specifically uses the sine function, which deals with ratios between angles and their opposite sides. For any angle \( \theta \) in a triangle, the sine function is given by:glyphosate\[\sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}}\]

Using Trigonometric Functions

• **Sine Function:** Useful in the Law of Sines for finding missing side lengths when paired with their opposite angles.
• **Calculator Use:** To effectively use trigonometric functions, employ a calculator to evaluate terms like \( \sin 23^{\circ} \) or \( \sin 110^{\circ} \).

These functions allow us to translate angles into applications that solve for unknowns, revealing the full scope of a triangle's dimensions.