Problem 14 - Precalculus with Limits

Question

Using the Law of Cosines, use the Law of Cosines to solve the triangle. Round your answers to two decimal places. $$ a=55, \quad b=25, \quad c=72 $$

Step-by-Step Solution

Answer

The angles of the triangle are \(A\), \(B\), and \(C\) - make sure to compute them and round off the answers to two decimal places.

Step 1: Understand the Law of Cosines

The Law of Cosines describes the relationship between the lengths of the sides of a triangle and the cosine of one of its angles. It can be expressed as \(c^2 = a^2 + b^2 - 2ab \cos(C)\), where \(C\) is the angle opposite side \(c\), and \(a\) and \(b\) are the lengths of the other two sides.

Step 2: Compute angle C

Arranging the Law of Cosines formula to solve for \(\cos(C)\), we get \(\cos(C) = (a^2 + b^2 - c^2) / 2ab = \((55^2 + 25^2 - 72^2) / (2 \times 55 \times 25)\). Using your calculator, find the arccosine or inverse cosine of \(\cos(C)\) to find \(C\) in degrees.

Step 3: Compute angle A

Similar to Step 2, we can find angle \(A\) using the Law of Cosines but with sides \(b\) and \(c\) and angle \(A\). So, \(\cos(A) = (b^2 + c^2 - a^2) / 2bc = (25^2 + 72^2 - 55^2) / (2 \times 25 \times 72)\). Find the inverse cosine to get \(A\) in degrees.

Step 4: Compute angle B

Once we have angles \(A\) and \(C\), we can easily compute for angle \(B\) using the sum of angles in a triangle, which is 180 degrees. So, \(B = 180 - A - C\).

Step 5: Round off your answers

Finally, round off your answers for angles \(A\), \(B\), and \(C\) to two decimal places as stated in the problem.

Key Concepts

Solving Triangles

Solving a triangle involves finding all the unknown sides and angles in a triangle based on the information provided. In this exercise, you've been given three sides of the triangle, specifically a=55, b=25, and c=72. This is commonly referred to as an 'SSS' (Side-Side-Side) situation. Here, you need to use the Law of Cosines to find all the angles. By applying the Law of Cosines, you can break down the problem into manageable steps.

• First, use the formula to find the cosine of one of the angles by rearranging it according to the given sides.
• Next, utilize a calculator to find the angle using inverse trigonometric functions like the arccosine.
• Repeat these steps to find the remaining angles.

This method ensures that you can solve for each angle accurately.

Trigonometry

Trigonometry deals with the study of relationships between the angles and sides of triangles. It is the basis for concepts like the Law of Cosines that we use when none of the angles is initially known. In trigonometry, understanding these relationships allows us to solve complex problems.
For our problem, the Law of Cosines works as a powerful tool when you can't directly apply simpler trigonometric laws like the Law of Sines.

• It allows the calculation of an angle when the three sides of the triangle are known.
• Helps in situations where the triangle is not a right triangle, and standard trigonometric ratios don't apply directly.

Grasping how the Law of Cosines functions bring clarity to solving various types of triangles.

Angle Calculations

Calculating angles in triangles using trigonometry can often be straightforward if approached methodically. The Law of Cosines provides the foundation for this by linking sides to angles. Here's how you can conduct these calculations effectively:

• The formula used is: \ \(c^2 = a^2 + b^2 - 2ab \cos(C)\ \).
• From this, the cosine of angle C can be isolated and calculated.
• Repeat for angles A and B using the appropriate formula plug-ins as shown in the exercise solutions.

Once the cosines are calculated, you can find the angles through inverse trigonometric functions like arccosine. It's crucial to ensure that all calculations follow logically and that the final angles, A, B, and C, when added, equal 180 degrees as per the sum of angles in a triangle. Always remember to round your final answers to two decimal places for precision.