Problem 31 - A Graphical Approach to Precalculus with Limits

Question

Solve triangle. \(A=39.70^{\circ}, C=30.35^{\circ}, b=39.74\) meters

Step-by-Step Solution

Answer

Angles: A = 39.70°, B = 109.95°, C = 30.35°; Sides: a ≈ 26.91 m, b = 39.74 m, c ≈ 21.20 m.

Step 1: Find Angle B

To find angle B in the triangle, use the property that the sum of all angles in a triangle equals 180°. Therefore, B = 180° - A - C. Calculate B as follows: \[ B = 180° - 39.70° - 30.35° = 109.95° \] So, angle B is 109.95°.

Step 2: Apply the Law of Sines

Use the Law of Sines to find other sides. The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We already know \( b = 39.74 \) meters and have the angles, so use this to find side \( a \): \[ \frac{a}{\sin 39.70°} = \frac{39.74}{\sin 109.95°} \]

Step 3: Solve for Side a

Rearrange the equation from Step 2 to solve for \( a \): \[ a = 39.74 \cdot \frac{\sin 39.70°}{\sin 109.95°} \] Calculate using a calculator: \[ a \approx 39.74 \cdot \frac{0.6393}{0.9455} \approx 26.91 \text{ meters} \]Thus, side \( a \) is approximately 26.91 meters.

Step 4: Solve for Side c Using the Law of Sines

Now, solve for side \( c \) using \( \frac{c}{\sin C} = \frac{b}{\sin B} \):\[ \frac{c}{\sin 30.35°} = \frac{39.74}{\sin 109.95°} \]

Step 5: Calculate Side c

Rearrange the equation to solve for \( c \):\[ c = 39.74 \cdot \frac{\sin 30.35°}{\sin 109.95°} \]Calculate using a calculator:\[ c \approx 39.74 \cdot \frac{0.5050}{0.9455} \approx 21.20 \text{ meters} \]Thus, side \( c \) is approximately 21.20 meters.

Key Concepts

Law of Sines

When solving triangles, the Law of Sines is a fundamental formula that helps us find unknown sides and angles. It connects the sides of a triangle to its angles, effectively bridging the gap between geometry and trigonometry.

The Law of Sines formula is expressed as:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

where:

• \( a, b, \) and \( c \) are the lengths of the sides of the triangle.
• \( A, B, \) and \( C \) are the angles opposite those respective sides.

In the exercise we're considering, we used this formula to find both side \( a \) and side \( c \) after determining angle \( B \).

A practical way to use the Law of Sines is to set up a proportion using known values. For instance, in our case, where \( b = 39.74 \) meters and \( B = 109.95^{\circ} \), the proportion helped us find unknown sides by leveraging the angles \( A \) and \( C \). This showcases the importance of having at least one side and opposing angle known before applying the law effectively.

Angle Calculation

The calculation of angles within a triangle is often the first step when solving for unknowns. It's vital to know that in any triangle, the sum of the three interior angles is always 180 degrees. This intrinsic property serves as our starting point.

To calculate an unknown angle, label the existing angles and solve the equation: \( A + B + C = 180^{\circ} \). By subtracting the sum of the known angles from 180 degrees, we find the unknown angle.

In our specific problem, the known angles were \( A = 39.70^{\circ} \) and \( C = 30.35^{\circ} \). So, to find angle B, we calculated:
\[B = 180^{\circ} - 39.70^{\circ} - 30.35^{\circ}\]
This calculation revealed that \( B = 109.95^{\circ} \), which was crucial for using the Law of Sines later on. It's this initial process of calculating angles that sets the foundation for further solving, as it provides the angles necessary for trigonometric identities and relationships in the triangle.

Triangle Properties

A triangle is a polygon with three edges and three vertices. Understanding its properties helps solve triangles effectively. The specific type of triangle we're dealing with is a non-right triangle with given angles and a side.

Key properties of triangles include:

• The angle sum property: the interior angles of any triangle add up to 180 degrees.
• Types of triangles based on angles and sides: Equilateral, Isosceles, and Scalene.
• Congruency and similarity rules: These dictate when triangles match exactly or have the same shape with different sizes.

The problem at hand had given angles and one side, allowing the use of
The Law of Sines to compute the missing dimensions.

By understanding properties such as angle relationships and the classification of triangles, we can readily grasp which methods to apply. This knowledge facilitates logical triangulation from angles to sides and vice versa, making the process of solving triangles smooth and intuitive.