Problem 16 - Precalculus with Calculus Previews

Question

Use the Law of Cosines to solve the triangle. $$ \beta=100^{\circ}, a=22.3, b=16.1 $$

Step-by-Step Solution

Answer

Side \( c \) is approximately 29.68.

Step 1: Identify Known Elements

From the given problem, we know the following elements of the triangle: the angle \( \beta = 100^\circ \), side \( a = 22.3 \), and side \( b = 16.1 \). We need to find side \( c \) using the Law of Cosines.

Step 2: Apply the Law of Cosines

The Law of Cosines states: \( c^2 = a^2 + b^2 - 2ab\cos(\beta) \). Substitute the known values into the formula: \[ c^2 = 22.3^2 + 16.1^2 - 2 \times 22.3 \times 16.1 \times \cos(100^\circ) \].

Step 3: Calculate \( a^2 \) and \( b^2 \)

Calculate the squares of \( a \) and \( b \): \( 22.3^2 = 497.29 \) and \( 16.1^2 = 259.21 \). Now, substitute these into the Law of Cosines: \[ c^2 = 497.29 + 259.21 - 2 \times 22.3 \times 16.1 \times \cos(100^\circ) \].

Step 4: Determine \( \cos(100^\circ) \) and Compute

Using a calculator, find \( \cos(100^\circ) \approx -0.1736 \). Substitute to get: \[ c^2 = 497.29 + 259.21 + 2 \times 22.3 \times 16.1 \times 0.1736 \]. Compute: \[ c^2 = 497.29 + 259.21 + 124.888 \approx 881.388 \].

Step 5: Solve for \( c \) by Taking Square Root

Take the square root of \( c^2 \): \( c = \sqrt{881.388} \approx 29.68 \). Therefore, \( c \approx 29.68 \).

Key Concepts

Solving Triangles

Solving a triangle means determining all three sides and all three angles of that triangle. When you have a mix of known angles and sides, you'll need to use trigonometry, specifically the Law of Cosines for certain cases.
This method is especially useful when you have either:

• Two sides and the included angle (SAS)
• All three sides known (SSS)

By applying the Law of Cosines, you can find the unknown sides or angles, which completes the data needed for the triangle.
In our exercise, we have two sides and an included angle, making it a perfect candidate for the Law of Cosines. Once you've found the missing side, you can proceed to find the missing angles using either the Law of Sines or by continuing with the Law of Cosines.
Remember, solving the triangle fully requires the lengths of all sides and the measures of all angles.

Angle-Side Relationships

Understanding the relationships between angles and sides is crucial in trigonometry.
The Law of Cosines not only helps us find missing sides but also shows how the angles relate to those sides.
For example, in any triangle, there's a relationship:

• The longest side is opposite the largest angle.
• The shortest side is opposite the smallest angle.

This relationship is pivotal when solving triangles as it helps in checking if the calculated answers make sense.
In our exercise, checking if the calculated angle fits this pattern is essential. If it doesn’t, it might indicate a calculation error.
Being aware of these relationships helps confirm that your triangle's elements are consistent with the fundamental properties of triangles.

Trigonometry

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles.
It's a powerful tool that allows us to solve for unknown measurements given limited information. The Law of Cosines is a fundamental part of trigonometry.
When using trigonometry:

• Start by identifying known values.
• Choose the appropriate formula based on the given information.
• Substitute values and solve systematically.

Trigonometry isn't just limited to right triangles, even though many might initially learn it this way.
The Law of Cosines expands its usefulness to any triangle, allowing you to work with obtuse and acute angles alike.
By mastering trigonometry, especially the Law of Cosines, students gain a deeper understanding of geometry and how diverse forms of triangles behave.