Problem 4 - Precalculus : Building Concepts and Connections

Question

Use the Law of Cosines to find the umknoten side of each of the given triangles. The standard notation for labeling of triangles is used. Round anstoers to four decimal places. $$a=10, b=5, C=102^{\circ}$$

Step-by-Step Solution

Answer

The solution to this problem is \(c = 13.4259\), rounded to four decimal places.

Step 1: Arrange equation

The first step is to arrange the Law of Cosines to solve for the unknown side \(c\). For this, arrange the law like so: \(c = \sqrt{a^2 + b^2 - 2ab \cos(C)}\)

Step 2: Substitute the given Values

Substitute the given values to the arranged equation. So, \(c = \sqrt{10^2 + 5^2 - 2 \cdot 10 \cdot 5 \cos(102^\circ)}\)

Step 3: Solve the Expression

Next, you need to calculate the expression inside the square root, which includes the cosine of the given angle. Remember to use degree mode in your calculator when finding the cosine since our angle is given in degrees. After that, find the square root to get the answer for \(c\).

Step 4: Rounding

Round off the answer to four decimal places as per the given instruction.

Key Concepts

Triangle Side Computation Using the Law of Cosines

The Law of Cosines is a fundamental relation in trigonometry that allows us to find an unknown side of a triangle when we are given two sides and the included angle. This is particularly useful when the triangle is not a right triangle, where the Pythagorean theorem can't be applied. It is given by:

• \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

Here, \( a \), \( b \), and \( c \) are the sides of the triangle, and \( C \) is the angle opposite the side \( c \).
To find the unknown side \( c \), rearrange the formula to solve for \( c \):

• \( c = \sqrt{a^2 + b^2 - 2ab \cos(C)} \)

Now, let's break down the process:1. **Input the given values**: Make sure to identify which side is which. In this problem, \( a = 10 \), \( b = 5 \), and \( C = 102^\circ \).
2. **Calculate**: Substitute these values into your rearranged formula to find \( c \). Perform all arithmetic operations step by step for accuracy.Once you have solved the equation, remember to take the square root to find the length of side \( c \). Careful calculation at each stage ensures that rounding errors are minimized.

Angle Measurement in Degrees

Angle measurements can be expressed in degrees or radians, but in many geometric calculations, such as using the Law of Cosines, degrees are often used. This is important as trigonometric functions like cosine need to know which mode (degrees or radians) they should be working in.When dealing with degrees:

• Make sure your calculator is set to "degree" mode. Most scientific calculators have this option; it's crucial not to overlook this step as it will affect all trigonometric calculations.

Understanding degrees:- A full circle is \( 360^\circ \). This means any angle is a fraction of a circle.- For instance, \( 90^\circ \) is a quarter of a full turn.The given angle in our problem is \( 102^\circ \), which is slightly larger than a right angle (\( 90^\circ \)).
Being familiar with how angles relate to each other will help you visualize and comprehend the geometry in trigonometric contexts. Adjusting your calculator settings each time you switch between degrees and radians is an essential skill.

Understanding Trigonometric Functions

Trigonometric functions are mathematical tools that relate the angles of a triangle to its sides. The three primary trigonometric functions are sine, cosine, and tangent, which are used frequently to solve various problems in geometry and trigonometry.

• **Cosine**: In our exercise, the cosine function helps determine the relationship between the sides and angle in a triangle. It is expressed as \( \cos(\theta) \), where \( \theta \) is the angle.

Why is cosine important here?- **Calculation**: It allows us to calculate the effect of an angle on the lengths of the sides it connects.
For instance, in our scenario with side \( c \):

• The cosine function helps reduce the influence of angle \( C \) on the length of side \( c \) based on the characteristics of the triangle.

The values of the cosine function change as the angle increases or decreases, ranging from -1 to 1. This range influences how much the angle will impact the sides, which is pivotal in precise calculations. Ensure you are comfortable with calculating and interpreting cosine values, as they are integral to trigonometry.