Problem 21 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Law of cosines to solve the triangle. $$B=8^{\circ} 15^{\prime}, \quad a=26, \quad c=18$$

Step-by-Step Solution

Answer

After calculating, we get angle \(B=8.25^{\circ}\), side \(b\) value calculated from step 2, angle \(A\) from step 3, and angle \(C\) from step 4. These values will completely solve the triangle.

Step 1: Convert Degrees and Minutes into Decimal Degrees

First, convert the angle B given in degrees and minutes into decimal degrees. Since 1 degree is equivalent to 60 minutes, we can convert 15 minutes into degrees by dividing it by 60. So, \( B = 8^{\circ} + \frac{15}{60}^{\circ} = 8.25^{\circ} \).

Step 2: Compute the third side (b)

Using the Law of Cosines, you can solve for the missing side \(b\), which is opposite to angle B. The Law of Cosines formula is \(b^2 = a^2 + c^2 - 2ac\cos(B)\). Substitute the given values into this formula to get \(b^2 = 26^2 + 18^2 - 2(26)(18)\cos(8.25)\). Solve this equation to find the value of \(b\).

Step 3: Determine Angle A

After finding the value of side \(b\), the Law of Cosines can be used again to find one of the remaining angles. Let's start with angle \(A\). The formula becomes: \(\cos(A) = \frac{b^2 + c^2 - a^2}{2bc}\). Substitute the known values into this formula and solve for \(A\).

Step 4: Determine Angle C

After finding angle A, the remaining angle, C, can be found by using the fact that the sum of all angles in a triangle should equal 180 degrees. Hence, \(C = 180^\circ - A - B\). Substitute the known values into the formula to solve for angle C.

Key Concepts

Triangle Solving

To solve a triangle means to find all its unknown sides and angles. When you are given certain pieces of information, like some angles and sides, you can use trigonometric formulas such as the Law of Cosines to find the missing parts.

• The Law of Cosines is particularly useful in situations where you know two sides and the included angle (SAS condition) or three sides (SSS condition).
• In this exercise, we're given one angle and two sides, so we begin by using the Law of Cosines to find the third side.

By systematically applying trigonometric laws, and formulas, we break down the task into smaller, manageable parts. This approach helps ensure accuracy and consistency in solving the triangle.

Angle Conversion

Angles can be measured in degrees and minutes, which often requires conversion for calculations. In this exercise, angle B is given in degrees and minutes.

• To convert minutes to degrees, you use the relationship that 1 degree equals 60 minutes.
• This conversion is crucial for using trigonometric functions and formulas correctly, as they typically require angles in decimal degrees.

For instance, converting the angle 8 degrees 15 minutes to decimal degrees, you divide 15 by 60 and add it to 8, resulting in 8.25 degrees. This step is essential for ensuring that your calculations with trigonometric functions, like cosine, are accurate.

Trigonometry

Trigonometry is the study of triangles, specifically the relationships between their angles and sides. It relies on functions like sine, cosine, and tangent to compute these relationships.

• The Law of Cosines is a fundamental trigonometric equation that helps find missing components in a triangle.
• In this case, we used it to compute the length of side b and later to find angle A.

Understanding how to apply these functions and formulas is a critical part of solving triangle problems and demonstrates the power of trigonometry in practical scenarios.

Angle Sum Property

The Angle Sum Property is a basic geometric principle stating that the sum of the angles in a triangle always equals 180 degrees.

• This property is essential when you have already found two angles, as you can easily calculate the third one.
• Once you have angles A and B, you can find angle C by subtracting these from 180 degrees.

For solving triangles, this property simplifies finding unknown angles after using trigonometric formulas to find others. It assures that all angles will accurately complete the triangle.