Problem 7 - Precalculus with Calculus Previews

Question

Use the Law of Cosines to solve the triangle. $$ a=11, b=9 \cdot 5, c=8.2 $$

Step-by-Step Solution

Answer

The angle opposite side \(c\) is approximately \(49.6^\circ\).

Step 1: Write the Law of Cosines Formula

The Law of Cosines states that for any triangle with sides \(a\), \(b\), and \(c\), and the angle opposite to the side \(c\) is \(C\), the formula is: \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\).

Step 2: Substitute Known Values into the Formula

Substitute the given side lengths into the Law of Cosines formula: \(8.2^2 = 11^2 + (9.5)^2 - 2 \times 11 \times 9.5 \times \cos(C)\).

Step 3: Calculate the Squares of Each Side Length

Calculate each square: \(8.2^2 = 67.24\), \(11^2 = 121\), and \((9.5)^2 = 90.25\).

Step 4: Plug Squared Values into the Equation

Replace squared values back into the equation: \(67.24 = 121 + 90.25 - 209 \cdot \cos(C)\).

Step 5: Simplify the Equation

Simplify the right side of the equation: \(67.24 = 211.25 - 209 \cdot \cos(C)\).

Step 6: Solve for \(\cos(C)\)

Rearrange the equation to find \(\cos(C)\): \(209 \cdot \cos(C) = 211.25 - 67.24\). Simplify to \(209 \cdot \cos(C) = 144.01\).

Step 7: Calculate \(\cos(C)\)

Divide both sides by 209 to isolate \(\cos(C)\): \(\cos(C) = \frac{144.01}{209}\).

Step 8: Find the Angle \(C\)

Use a calculator to find the angle \(C\) by computing \(\cos^{-1}\left(\frac{144.01}{209}\right)\). Use the calculator to determine \(C\approx 49.6^\circ\).

Key Concepts

Solving Triangles

Solving a triangle involves finding unknown sides or angles given some initial information, like side lengths or angles. This process helps understand how different elements of a triangle are related. It can be especially useful in fields such as surveying, architecture, and physics. To solve a triangle, you can use various methods, one of the most important being the Law of Cosines when dealing with any triangle (not just right triangles).
In this specific exercise, with sides given as \(a=11\), \(b=9.5\), and \(c=8.2\), it requires finding the unknown angle opposite side \(c\). This approach is helpful in real-world problems where sometimes only side lengths are accessible.
To start solving such a triangle, choose a method based on the given information:

• Use the Law of Cosines for non-right triangles when given two sides and an included angle, or three sides and need to find an angle.
• Use the Law of Sines when you know either two angles and a side or two sides and a non-included angle.

Trigonometry

Trigonometry is the branch of mathematics that studies relationships between the sides and angles of triangles. To solve triangles efficiently, understanding trigonometric concepts such as sine, cosine, and tangent is crucial.
The Law of Cosines, an essential tool in trigonometry, extends the Pythagorean theorem for non-right triangles. This law states that for any triangle, the square of one side equals the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle. Symbolically, this can be written as:

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

When two sides and the included angle of a triangle are known, you can apply this formula to find the third side. Conversely, if all three sides are known, as in this problem, it allows you to compute any of the triangle's angles.
Through trigonometry, we glean how angles more than just link side lengths; they add depth to understand the shape and size of the spaces we examine.

Angle Calculation

Calculating angles in a triangle involves using side lengths alongside trigonometric formulas. In this exercise, we used the Law of Cosines to calculate angle \(C\) opposite to side \(c\).
Here's a step-by-step guide to finding an angle using the Law of Cosines:

• First, substitute the known side lengths into the Law of Cosines formula: \(8.2^2 = 11^2 + (9.5)^2 - 2 \times 11 \times 9.5 \times \cos(C)\).
• Calculate the squares of each side: \(8.2^2 = 67.24\), \(11^2 = 121\), and \(9.5^2 = 90.25\).
• Replace these values in the equation and simplify to find \(209 \cdot \cos(C) = 144.01\).
• Then solve for \(\cos(C)\) by dividing \(144.01\) by \(209\) to isolate \(\cos(C)\).
• Finally, use a calculator for \(\cos^{-1}(\text{value})\) to compute the actual angle measure. For this problem, \(C\) was approximately \(49.6^\circ\).

Using these steps, one can calculate any missing angle in a triangle if the side lengths are known, showcasing how powerful trigonometry can be in uncovering critical geometric properties.