Problem 13 - 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. $$ a=11, \quad b=15, \quad c=21 $$

Step-by-Step Solution

Answer

The solutions are \(A = \cos^{-1}(\frac{{15^2 + 21^2 - 11^2}}{{2*15*21}})\), \(B = \cos^{-1}(\frac{{11^2 + 21^2 - 15^2}}{{2*11*21}})\), and \(C = 180 - A - B\). Each of these are rounded to two decimal places.

Step 1: Calculate Angle A

Use the law of Cosines to solve for angle A. The formula for angle A, \(\cos(A) = \frac{{b^2 + c^2 - a^2}}{{2bc}}\), becomes \(\cos(A) = \frac{{15^2 + 21^2 - 11^2}}{{2*15*21}}\)

Step 2: Find the Actual Angle A

Use the inverse cosine function to find the actual angle A in degrees: \(A = \cos^{-1}(\frac{{15^2 + 21^2 - 11^2}}{{2*15*21}})\)

Step 3: Calculate Angle B

Apply the law of Cosines to solve for angle B. The formula for angle B, \(\cos(B) = \frac{{a^2 + c^2 - b^2}}{{2ac}}\), becomes \(\cos(B) = \frac{{11^2 + 21^2 - 15^2}}{{2*11*21}}\)

Step 4: Find the Actual Angle B

Use the inverse cosine function to find the actual angle B in degrees: \(B = \cos^{-1}(\frac{{11^2 + 21^2 - 15^2}}{{2*11*21}})\)

Step 5: Use the Triangle Sum Property

Now that we have two angles, we can find the third by using the Triangle Sum Property. It states that the sum of the angles in a triangle is 180 degrees: \(C = 180 - A - B\)

Step 6: Round your answers

Round each of your final answers to two decimal places

Key Concepts

Triangle Sum Property

The Triangle Sum Property is a fundamental rule in geometry. This property states that the sum of the interior angles of any triangle is always 180 degrees. This knowledge is useful for solving a triangle when two angles are known.
Suppose you have a triangle with angles A, B, and C. According to the Triangle Sum Property:

• Angle A + Angle B + Angle C = 180°

To find the third angle when two are known, you simply subtract the sum of the known angles from 180°:

• Angle C = 180° - (Angle A + Angle B)

This property is incredibly helpful as it allows you to fully determine the triangle. Knowing this can speed up solving problems, especially in a situation where the Law of Cosines is used to find the first two angles.

Cosine Rule

The Cosine Rule, also known as the Law of Cosines, is an essential tool for solving triangles, especially non-right angled ones. It allows you to calculate an unknown side or angle when you know a little bit about the other parts of the triangle.
The formula for the cosine of angle A is:

• \( \cos(A) = \frac{{b^2 + c^2 - a^2}}{{2bc}} \)

You can similarly use formulas to find cosines of angles B and C if needed:

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

These formulas help relate the sides to the angles, making it possible to solve for unknown components. The Law of Cosines is particularly useful when you need to find the angle opposite to the longest side in the triangle.

Inverse Cosine Function

The inverse cosine function helps you find the angle when you have the cosine of that angle. In trigonometry, the inverse cosine function is denoted as \( \cos^{-1} \). This function will return the angle whose cosine is the given number.
For example, if you know \( \cos(A) = x \), then:

• \( A = \cos^{-1}(x) \)

This transformation from a cosine value back to an angle is crucial because the Law of Cosines typically calculates the cosine of angles for us, not the angles themselves.
When using trigonometric functions on a calculator, make sure it is set to the correct mode (degrees or radians) to obtain the correct measure of the angle. The inverse cosine function is foundational for converting side-based calculations into angles, thereby completing the puzzle of a triangle's dimensions.