Problem 63 - Trigonometry

Question

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Find \(c\) for triangle \(A B C\) if \(a=6.8\) meters, \(b=8.4\) meters, and \(C=48^{\circ}\). a. \(5.6 \mathrm{~m}\) b. \(40 \mathrm{~m}\) c. \(8.6 \mathrm{~m}\) d. \(6.4 \mathrm{~m}\)

Step-by-Step Solution

Answer

The length of side \( c \) is approximately 6.4 m, option d.

Step 1: Identify the Triangle Law to Use

In this exercise, we need to find the length of side \( c \) in triangle \( ABC \). It is given that \( a = 6.8 \) m, \( b = 8.4 \) m, and \( C = 48^{\circ} \). The law of cosines is suitable for finding a side in a triangle when two sides and the included angle are known.

Step 2: Write the Law of Cosines Formula

The law of cosines formula is \( c^2 = a^2 + b^2 - 2ab \cos(C) \). Here \( C \) is the angle opposite side \( c \). This formula will allow us to calculate the length of side \( c \).

Step 3: Substitute the Known Values

Substitute the known values into the law of cosines formula: \( c^2 = (6.8)^2 + (8.4)^2 - 2 \times 6.8 \times 8.4 \times \cos(48^{\circ}) \).

Step 4: Calculate the Squares of Known Sides

Calculate \((6.8)^2 = 46.24\) and \((8.4)^2 = 70.56\). The equation becomes \( c^2 = 46.24 + 70.56 - 2 \times 6.8 \times 8.4 \times \cos(48^{\circ}) \).

Step 5: Apply Cosine Function

Calculate \(2 \times 6.8 \times 8.4 = 114.24\) and then find \( \cos(48^{\circ}) \). Use a calculator to find that \( \cos(48^{\circ}) \approx 0.6691 \).

Step 6: Complete the Calculation

Substitute \( \cos(48^{\circ}) \approx 0.6691 \) into the equation: \( c^2 = 116.8 - 114.24 \times 0.6691 \). Calculate the product: \( 114.24 \times 0.6691 = 76.439584 \) and subtract it from \( 116.8 \) to find \( c^2 = 40.360416 \).

Step 7: Find the Square Root

To find \( c \), take the square root of \( 40.360416 \): \( c \approx \sqrt{40.360416} \). This calculation results in \( c \approx 6.4 \) meters.

Key Concepts

Triangle Side Calculation

Understanding how to calculate the side of a triangle is crucial, particularly when dealing with non-right triangles where simple Pythagorean Theorem applications fall short. The Law of Cosines is a powerful tool in these cases. It allows us to find a missing side when we know two sides and the included angle. For a triangle where the sides are labeled as \( a \), \( b \), and \( c \), and the angle opposite side \( c \) is labeled \( C \), the Law of Cosines formula is:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
To find the length of side \( c \), follow these steps:

• Substitute the known values of sides \( a \), \( b \), and angle \( C \) into the formula.
• Calculate the squares of the lengths of \( a \) and \( b \).
• Multiply \( 2 \times a \times b \times \cos(C) \) and substitute these values back into the equation.
• Solve for \( c^2 \) and find \( c \) by taking the square root.

This step-by-step approach simplifies problems that involve oblique triangles.

Included Angle

The included angle in a triangle is the angle that directly determines the triangle's other dimensions when combined with given side lengths. In our example, angle \( C \) is the included angle between sides \( a \) and \( b \) in triangle \( ABC\). It plays a pivotal role in calculations using the Law of Cosines.
Here’s why it matters:

• It helps in establishing the relationship between one of the sides being calculated and the sum of the squared lengths of the other two sides.
• In the Law of Cosines, \( \cos(C) \) specifically affects the result of \( c^2 \), which means even small changes in \( C \) can lead to different \( c \) values.

When solving for side lengths or proving triangle properties, understanding which angle is included can significantly enhance precision and comprehension.

Trigonometric Functions

Trigonometric functions, especially the cosine function, are central in the context of the Law of Cosines. For a triangle, \( \cos(\) of an angle connects the angle to side lengths. It specifies how the angular opening influences the extent of the opposite side.
Here's how the cosine function is applied in the given solution:

• The cosine of the included angle \( C \) regulates the deductive term in \( c^2 = a^2 + b^2 - 2ab \cos(C) \).
• The value of \( \cos(48^{\circ}) \) is approximately 0.6691, derived using a calculator, which can then be implemented in further calculations.

This function reduces how much the two sides \( a \) and \( b \) are squared and summed up to determine \( c \. \), showing how geometric angles relate to side lengths mathematically.