Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the Law of sines or the Law of cosines can be used to find another measure of the triangle. Then solve the triangle. $$A=24^{\circ}, \quad a=4, \quad b=18$$

Step-by-Step Solution

Answer

The measures of the triangle are: \( A = 24^{\circ} \), \( B \approx 78.46^{\circ} \), \( C \approx 77.54^{\circ} \), \( a = 4 \), \( b = 18 \), and \( c \approx 18.52 \).

Step 1: Understand the given values

Given values are: \( A = 24^{\circ} \), \( a = 4 \), and \( b = 18 \).

Step 2: Apply the Law of Sines to find angle B

The Law of Sines stipulates that the ratio of the length of a side in a triangle to the sine of the angle opposite that side is the same for all three sides of the triangle. Using this law, it is possible to solve for angle B using the formula: \( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} \). Rearrange this equation to solve for \( \sin(B) \) to get: \( \sin(B) = \frac{b \sin(A)}{a} \). After inserting the given values and calculating, we find that \( B \approx 78.46^{\circ} \).

Step 3: Use the triangle angle sum to find angle C

The sum of the angles in a triangle always equals \(180^{\circ}\). So, \( C = 180^{\circ} - A - B \). With A and B already known, calculating gives \( C \approx 77.54^{\circ} \)

Step 4: Apply the Law of Sines to find side c

Using the formula \( \frac{a}{\sin(A)} = \frac{c}{\sin(C)} \), we can solve for side c. Rearrange the equation to \( c = \frac{a \sin(C)}{\sin(A)} \), plug in the known values, and calculate to find \( c \approx 18.52 \).

Key Concepts

Law of Sines

The Law of Sines is a fundamental concept in trigonometry used to solve triangles, especially in finding unknown angles or sides in oblique triangles. An oblique triangle is one without a right angle. The law states that the ratio between the length of a side and the sine of its opposing angle is constant for all three sides in a triangle.

In mathematical terms, it's expressed as:

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

This means you can find a missing angle or side if you know at least one pair of side and opposing angle, plus one additional measurement (another angle or side).

In the original exercise, the Law of Sines was used effectively to calculate the unknown angle \( B \), using the known side \( b \) and its opposing angle \( A \), leading to the relationship \( \sin(B) = \frac{b \sin(A)}{a} \). Once you have \( \sin(B) \), you can find angle \( B \) easily. This approach demonstrates the powerful utility of the Law of Sines in solving triangles.

Law of Cosines

The Law of Cosines extends the possibilities of solving triangles beyond what the Law of Sines offers. It is particularly useful when you know two sides of a triangle and the included angle or when all three sides are known and need to find an unknown angle. It bridges the gap when the Law of Sines cannot be applied due to the triangle's specific known values.

The Law of Cosines is expressed as:

• \( c^2 = a^2 + b^2 - 2ab \cos(C) \)
• \( b^2 = a^2 + c^2 - 2ac \cos(B) \)
• \( a^2 = b^2 + c^2 - 2bc \cos(A) \)

This equation is similar to the Pythagorean theorem with an additional term to account for the angle between the sides. Notably in triangles where neither angle nor opposite side is known together, the Law of Cosines can become quite handy.

Though it wasn't used directly to solve the original exercise, it's essential understanding it helps solve many similar trigonometry problems.

Triangle Angle Sum

The Triangle Angle Sum is a simple yet crucial property of triangles that states the sum of all internal angles in any triangle is always \(180^{\circ}\). This fundamental property is a cornerstone in geometry and is particularly helpful for determining unknown angles when at least two angles in the triangle are known.

The formula is straightforward:

• \( A + B + C = 180^{\circ} \)

This relationship comes into play in the original exercise when needing to find the remaining angle \( C \) after calculating angles \( A \) and \( B \). Once the two angles are known through calculations or given data, finding the third angle is a simple subtraction problem, e.g., \( C = 180^{\circ} - A - B \).

Understanding this property will often be the final step in solving a triangle, ensuring that you have consistent and valid angle measures, reinforcing accurate problem-solving across various trigonometric challenges.