Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Solve each triangle. \(C=45.6^{\circ}, b=8.94\) meters, \(a=7.23\) meters

Step-by-Step Solution

Answer

Side \( c \) and angles \( A \) and \( B \) can be determined using the Law of Cosines and the Law of Sines.

Step 1: Use the Law of Cosines to find Side c

Since we have two sides and the included angle, we can use the Law of Cosines to find the third side, \( c \). The Law of Cosines is given by \[ c^2 = a^2 + b^2 - 2ab \cos(C) \]Substituting the values, \[ c^2 = 7.23^2 + 8.94^2 - 2(7.23)(8.94) \cos(45.6^{\circ}) \]Calculate \( c^2 \) and then take the square root to find \( c \).

Step 2: Calculate Angle A or B using the Law of Sines

We can now use the Law of Sines to find one of the remaining angles. Let's find angle \( A \) using the formula:\[ \frac{a}{\sin A} = \frac{c}{\sin C} \]Rearrange to find \( \sin A \):\[ \sin A = \frac{a \sin C}{c} \]Substitute known values and solve for \( A \) using \( \sin^{-1} \).

Step 3: Calculate the Third Angle

Now that we have angles \( C \) and \( A \), we can find angle \( B \) using the triangle sum theorem:\[ A + B + C = 180^{\circ} \]Rearrange to solve for \( B \):\[ B = 180^{\circ} - A - C \]Substitute the known values of \( A \) and \( C \) to find \( B \).

Key Concepts

Law of Cosines

The Law of Cosines is an essential mathematical formula used to determine a side or angle of a triangle when certain other sides and angles are known. This rule extends the Pythagorean theorem, making it applicable to all types of triangles, not just right-angled ones. It is particularly useful when dealing with non-right triangles where you know the values of two sides and the included angle. In mathematical terms, the law is expressed as:

• \( c^2 = a^2 + b^2 - 2ab \cos(C) \)

Here, \( c \) represents the unknown side, while \( a \) and \( b \) are the known sides and \( C \) is the angle opposite to side \( c \). To find \( c \), you substitute the values of \( a \), \( b \), and \( C \) into the formula, simplify, and finally take the square root. This approach helps in deducing the third side, even when the triangle does not have a right angle. This is often the first step toward solving the triangle, enabling subsequent calculations to determine the remaining angles.

Law of Sines

The Law of Sines is a powerful rule used in trigonometry to find unknown angles or sides in a triangle. Specifically, it is used when you have information about two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). The relationship is given by the formula:

• \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)

In this equation, \( a \), \( b \), and \( c \) are the lengths of the sides opposite to angles \( A \), \( B \), and \( C \) respectively. It states that the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. When solving a triangle, if you know two sides and an angle, as in the problem, you can rearrange the formula to solve for one of the missing angles. This approach is intuitive once you understand that angles and their respective sides have proportional relationships, which is the essence of the Law of Sines.

Triangle Sum Theorem

The Triangle Sum Theorem is a fundamental principle in geometry which asserts that the sum of the internal angles of a triangle is always \( 180^{\circ} \). This theorem forms the basis for finding the third angle of a triangle when the other two angles are known. By subtracting the sum of the known angles from \( 180^{\circ} \), you can easily calculate the value of the third angle:

• \( A + B + C = 180^{\circ} \)
• \( B = 180^{\circ} - A - C \)

In this way, the Triangle Sum Theorem guarantees that no matter the type of triangle, be it acute, obtuse, or right, the internal angles will always add up to \( 180^{\circ} \). This is particularly useful when solving triangles as it provides the final piece of the angle puzzle, ensuring that once you know any two angles, you can readily calculate the third.