Problem 1 - Contemporary Precalculus

Question

Directions: Standard notation for triangle \(A B C\) is used throughout. Use a calculator and round off your answers to one decimal place at the end of the computation. Solve the triangle ABC under the given conditions. $$A=40^{\circ}, b=10, c=7$$

Step-by-Step Solution

Answer

Question: Given a triangle ABC with angle A = \(40^{\circ}\), side b = 10, and side c = 7, find the length of side a and the measures of angles B and C. Round your answers to one decimal place. Answer: In triangle ABC, side a ≈ 5.1, angle B ≈ \(71.5^{\circ}\) and angle C ≈ \(68.5^{\circ}\).

Step 1: Identify the given information

In triangle ABC, we are given the following information: - Angle A = \(40^{\circ}\) - Side b = 10 - Side c = 7 We need to find the length of side a and the measures of angles B and C.

Step 2: Use Law of Cosines to find side a

To determine the length of side a, we can use the Law of Cosines: $$a^2 = b^2 + c^2 - 2 \cdot b \cdot c \cdot \cos {A}$$ Substitute the given values into the equation: $$a^2 = 10^2 + 7^2 - 2 \cdot 10 \cdot 7 \cdot \cos {40^{\circ}}$$

Step 3: Calculate side a length

Use a calculator to compute the value of side a: $$a^2 = 100 + 49 - 140 \cdot \cos{40^{\circ}}$$ After finding the value, we get \(a^2 \approx 26.43\). Now, take the square root of both sides to find the length of side a: $$a = \sqrt{26.43} \approx 5.1$$

Step 4: Use Law of Sines to find angle B

Now that we have the length of side a, we can use the Law of Sines to find angle B: $$\frac{\sin{B}}{b} = \frac{\sin{A}}{a}$$ Substitute the values we have found: $$\frac{\sin{B}}{10} = \frac{\sin{40^{\circ}}}{5.1}$$

Step 5: Calculate angle B measure

Multiply both sides by 10 and use a calculator to find angle B: $$\sin{B} = \frac{10 \cdot \sin{40^{\circ}}}{5.1}$$ Using a calculator, we find \(\sin{B} \approx 0.9512\). Now, find the inverse sine of 0.9512 to get the measure of angle B: $$B = \sin^{-1}{0.9512} \approx 71.5^{\circ}$$

Step 6: Calculate angle C measure

Since we now have the measures of angles A and B, we can use the sum of interior angles of a triangle to determine angle C: $$A + B + C = 180^{\circ}$$ Substitute the values of A and B: $$40^{\circ} + 71.5^{\circ} + C = 180^{\circ}$$ Then, we find the measure of angle C: $$C = 180^{\circ} - 40^{\circ} - 71.5^{\circ}$$ $$C \approx 68.5^{\circ}$$ The solved triangle ABC has the following measures: - Side a = 5.1 - Side b = 10 - Side c = 7 - Angle A = \(40^{\circ}\) - Angle B = \(71.5^{\circ}\) - Angle C = \(68.5^{\circ}\)

Key Concepts

Law of Cosines

Understanding the Law of Cosines is critical when solving triangles, especially when we are not dealing with a right-angled triangle. This law allows us to find the lengths of sides or the measure of angles when we know at least one angle and two sides. The Law of Cosines is represented by the formula:
c^2 = a^2 + b^2 - 2ab * cos(C)
It resembles the Pythagorean theorem but with an additional component that accounts for the included angle. For example, in our exercise where we know sides \(b=10\), \(c=7\), and angle \(A=40^{\textdegree}\), we were able to determine the length of side \(a\) by rearranging the Law of Cosines to
a^2 = b^2 + c^2 - 2bc * cos(A)
This formula becomes particularly useful in triangles where no side is perpendicular to another (non-right triangles). It's important to round only at the end of calculations to minimize error.

Law of Sines

The Law of Sines is an essential concept for solving for unknown angles and sides in a triangle, providing a relationship between the lengths of the sides and the sines of the angles. The formula is given by:
\frac{a}{\text{sin}(A)} = \frac{b}{\text{sin}(B)} = \frac{c}{\text{sin}(C)}
When we found the length of side \(a\) in our exercise, we were ready to use the Law of Sines to solve for angle \(B\). We set up the proportion
\frac{\text{sin}(B)}{b} = \frac{\text{sin}(A)}{a}
By substituting the known values, we could find \(\text{sin}(B)\) and then find the angle itself by using the inverse sine function. This law is particularly useful when the triangle does not contain a right angle, and we know at least one other angle and the length of the side opposite that angle.

Triangle Interior Angles

Triangles are unique figures in geometry, and one of their fundamental properties is that the sum of their interior angles always equals \(180^{\textdegree}\). This constant allows for a variety of problem-solving techniques. In the exercise, after finding angles \(A\) and \(B\), we were able to easily deduce angle \(C\) by subtracting the sum of angles \(A\) and \(B\) from \(180^{\textdegree}\). This property of triangle interior angles is vital and can be used to find missing angles, regardless of the triangle's type (acute, obtuse, or right). Hence:
C = 180^{\textdegree} - A - B
Understanding this relationship is crucial for solving triangles and is often used in conjunction with the Laws of Sines and Cosines for complete solutions.