Problem 17 - Algebra and Trigonometry

Question

In Problems 17-32, solve each triangle. $$ a=3, \quad b=4, \quad C=40^{\circ} $$

Step-by-Step Solution

Answer

The unknown side \(c \approx 2.57\), angle \(A \approx 48.6^{\circ}\), and angle \(B \approx 91.4^{\circ}\).

Step 1: Use the Law of Cosines

The Law of Cosines helps to find the unknown side when two sides and the included angle are known. The formula is given by: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \]Substitute the known values: \[ c^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \cos(40^{\circ}) \]Calculate: \[ c^2 = 9 + 16 - 24 \cdot \cos(40^{\circ}) \]\[ c^2 = 25 - 24 \cdot \cos(40^{\circ}) \]\[ c = \sqrt{25 - 24 \cdot \cos(40^{\circ})} \]\[ c \approx \sqrt{25 - 24 \cdot 0.766} \]\[ c \approx \sqrt{6.616} \]\[ c \approx 2.57 \]So, the unknown side \(c\) is approximately 2.57.

Step 2: Use the Law of Sines

Next, use the Law of Sines to find one of the remaining angles. The formula is: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]We will use \( \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \): \[ \frac{3}{\sin(A)} = \frac{2.57}{\sin(40^{\circ})} \]Solve for \(\sin(A)\): \[ \sin(A) = \frac{3 \cdot \sin(40^{\circ})}{2.57} \]Calculate: \[ \sin(A) = \frac{3 \cdot 0.6428}{2.57} \]\[ \sin(A) \approx 0.749 \]\( \sin^{-1}(0.749) \approx 48.6^{\circ} \)So, angle \(A\) is approximately \(48.6^{\circ}\).

Step 3: Find the Third Angle

The sum of angles in a triangle is always \(180^{\circ}\). Thus, to find angle \(B\): \[ B = 180^{\circ} - A - C \]Substitute the known values: \[ B = 180^{\circ} - 48.6^{\circ} - 40^{\circ} \]\[ B = 91.4^{\circ} \]So, angle \(B\) is approximately \(91.4^{\circ}\).

Key Concepts

Law of Cosines

The Law of Cosines is a useful tool for finding an unknown side of a triangle when you know two sides and the included angle. It is especially handy for non-right triangles. The formula is given by: \[ c^2 = a^2 + b^2 - 2ab \, \cos(C) \] In this exercise, we used the known values: \( a = 3 \), \( b = 4 \), and \( C = 40^{\circ} \). Plugging these values into the formula, we calculated:\[ c^2 = 3^2 + 4^2 - 2 \cdot 3 \cdot 4 \cdot \cos(40^{\circ}) \] This simplifies step-by-step to \( c \approx 2.57 \). Calculating each step can be done with a calculator, especially to handle the cosine of the angle.

Law of Sines

The Law of Sines relates the sides and angles of any triangle. This law is particularly useful when dealing with angles. The basic formula is: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \] To find one of the remaining angles, we selected \( \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \). Substituting known values, we calculated: \[ \frac{3}{\sin(A)} = \frac{2.57}{\sin(40^{\circ})} \] Solving for \( \sin(A) \, \text{we found that} \, \sin(A) \approx 0.749 \), which translates to an angle \( A \approx 48.6^{\circ} \). This process shows how we can find an unknown angle when certain sides and angles are known.

Triangle Angles

The sum of all interior angles in any triangle is \( 180^{\circ} \). When we know two angles, we can easily find the third by subtracting the sum of the two known angles from \( 180^{\circ} \). In this exercise, after finding one of the unknown angles (\( A \approx 48.6^{\circ} \)), we calculated the third angle \( B \) using the equation: \( B = 180^{\circ} - A - C \) After replacing \( A \) and \( C \) with the known values: \( 180^{\circ} - 48.6^{\circ} - 40^{\circ} \), we determined \( B \approx 91.4^{\circ} \). This method ensures that we always find the correct sum of triangle angles to be \( 180^{\circ} \).