Problem 13 - Algebra and Trigonometry

Question

Solve triangle \(A B C\). \(a=3.0, \quad b=4.0, \quad \angle C=53^{\circ}\)

Step-by-Step Solution

Answer

Side c ≈ 3.25; angles A ≈ 48.7° and B ≈ 78.3°.

Step 1: Understand the Law of Cosines

To solve the triangle ABC, where sides a and b and angle C are given, first use the Law of Cosines to find the unknown side c. The formula is: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]

Step 2: Calculate Side c

Substitute the known values into the Law of Cosines:\[ c^2 = 3.0^2 + 4.0^2 - 2 \cdot 3.0 \cdot 4.0 \cdot \cos(53^{\circ}) \]Calculate:\[ c^2 = 9 + 16 - 24 \cdot \cos(53^{\circ}) \]Find the cosine of 53 degrees and complete the calculation:\[ \cos(53^{\circ}) \approx 0.6018 \]\[ c^2 = 25 - 24 \cdot 0.6018 \]\[ c^2 = 25 - 14.4432 \]\[ c^2 \approx 10.5568 \]\[ c \approx \sqrt{10.5568} \approx 3.25 \]

Step 3: Use the Law of Sines to find Angle B

Now use the Law of Sines to find angle B:\[ \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]Rearrange to solve for \( \sin(B) \):\[ \sin(B) = \frac{b \cdot \sin(C)}{c} \]Substitute the values:\[ \sin(B) = \frac{4.0 \cdot \sin(53^{\circ})}{3.25} \]Calculating further:\[ \sin(B) = \frac{4.0 \cdot 0.7986}{3.25} \approx 0.982 \]Find \( B \) by taking the inverse sine:\[ B \approx \sin^{-1}(0.982) \approx 78.3^{\circ} \]

Step 4: Calculate Angle A

To find angle A, utilize the angle sum of a triangle:\[ A + B + C = 180^{\circ} \]\[ A = 180^{\circ} - 78.3^{\circ} - 53^{\circ} \]\[ A \approx 48.7^{\circ} \]

Step 5: Conclusion: Provide the Solution

The solved triangle has the following properties:Sides:- \( a = 3.0 \)- \( b = 4.0 \)- \( c \approx 3.25 \)Angles:- \( A \approx 48.7^{\circ} \)- \( B \approx 78.3^{\circ} \)- \( C = 53^{\circ} \)

Key Concepts

Law of Cosines

The Law of Cosines is a crucial tool in trigonometry, especially when dealing with non-right triangles. It is essential when you have two sides of a triangle and the included angle, and you want to find the third side. The formula for the Law of Cosines is:

• \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)

This formula resembles the Pythagorean theorem but includes an additional term that accounts for the triangle not being a right-angled one.
We use the cosine of angle \( C \) to adjust the calculation based on how different side \( c \) is from a right triangle’s hypotenuse.
To apply the Law of Cosines in the given problem where \( a = 3.0 \), \( b = 4.0 \), and \( \angle C = 53^\circ \), we plug these values into the formula and solve step by step.
After finding \( \cos(53^\circ) \approx 0.6018 \), the calculation simplifies to \( c^2 = 25 - 14.4432 \), so \( c \approx \sqrt{10.5568} \approx 3.25 \). This result helps us proceed to find other unknowns in the triangle.

Law of Sines

Once one side and its opposite angle are known, the Law of Sines becomes very helpful. This law states that:

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

This formula provides a way to calculate the unknown angles or sides when certain other values are known.
In the problem, after determining side \( c \) using the Law of Cosines, the Law of Sines helps find angle \( B \).
We rearrange the equation to solve for \( \sin(B) \):

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

Substituting in the known values, we find \( \sin(B) \approx 0.982 \). By taking the inverse sine, we find \( B \approx 78.3^\circ \). This method ensures we are building solutions using consistent set relations between angles and sides of triangles.

Triangle Solutions

Solving a triangle means finding all its sides and angles. Once one side and two angles or two sides and one angle are known, you can determine the missing parts using trigonometric laws.
For triangle \( ABC \) given \( a = 3.0 \), \( b = 4.0 \), and \( \angle C = 53^\circ \), we begin by finding side \( c \) using the Law of Cosines. Next, the Law of Sines helps us figure out angle \( B \).
To completely solve the triangle, remember that the sum of angles in a triangle is always \( 180^\circ \).

• \( A + B + C = 180^\circ \)

Thus, angle \( A \) is found by subtracting the sum of angles \( B \) and \( C \) from \( 180^\circ \).
Here, \( A \approx 180^\circ - 78.3^\circ - 53^\circ \approx 48.7^\circ \). Complete with this information, you can confidently state the complete triangle properties - all sides and angles.