Problem 8 - Precalculus: Functions and Graphs

Question

Approximate the remaining parts of triangle \(A B C\). $$a=37, \quad b=55, \quad c=43$$

Step-by-Step Solution

Answer

Approximate angles A, B, and C using the Law of Cosines and verify the results are consistent with triangle properties.

Step 1: Identify the Known Values

We know that Triangle ABC has sides of lengths \(a = 37\), \(b = 55\), and \(c = 43\). These are the side lengths of the triangle.

Step 2: Use the Law of Cosines to Find Angle A

To find angle A, we use the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab \cos(C) \). Substitute the known values: \( 43^2 = 37^2 + 55^2 - 2 \times 37 \times 55 \times \cos(A) \). By solving, \( 1849 = 1369 + 3025 - 4070\cos(A) \). Rearrange for \(\cos(A)\).

Step 3: Simplify to Find \(\cos(A)\)

Calculate \(1849 = 1369 + 3025 - 4070 \cos(A)\). This simplifies to \(1849 = 4394 - 4070 \cos(A)\), and then \(4070\cos(A) = 2545\). Compute \(\cos(A) = \frac{2545}{4070}\).

Step 4: Use Calculator to Find Angle A

Using a calculator, find \(A = \cos^{-1}(\frac{2545}{4070})\). This provides angle A in degrees.

Step 5: Use the Law of Cosines Again for Angle B

Use the Law of Cosines again for angle B: \( a^2 = b^2 + c^2 - 2bc \cos(B) \). Substitute: \( 37^2 = 55^2 + 43^2 - 2 \times 55 \times 43 \times \cos(B) \). Solve for \(\cos(B)\).

Step 6: Simplify to Find \(\cos(B)\)

Calculate \(1369 = 3025 + 1849 - 4730 \cos(B)\). This simplifies to \(1369 = 4874 - 4730 \cos(B)\), and then \(4730 \cos(B) = 3505\). Compute \(\cos(B) = \frac{3505}{4730}\).

Step 7: Use Calculator to Find Angle B

Using a calculator, find \(B = \cos^{-1}(\frac{3505}{4730})\). This provides angle B in degrees.

Step 8: Calculate Angle C

Since the sum of angles in a triangle is 180 degrees, calculate \(C = 180^\circ - A^\circ - B^\circ\) using the previously found values of A and B.

Step 9: Verify the Triangle

Check if all calculated angles (A, B, and C) add up to 180 degrees. Ensure precision in calculations.

Key Concepts

Triangle

Triangles are fundamental shapes in geometry, consisting of three sides, three angles, and three vertices. The sides are connected by straight lines and the connection of sides forms the triangle's angles. In any triangle, the sum of the interior angles is always 180 degrees. This is a critical concept because it helps to find a missing angle if the other two angles are known.

Triangles can be classified into different types based on their sides and angles. For instance:

• An **equilateral triangle** has all sides equal and all angles equal to 60 degrees.
• An **isosceles triangle** has two sides equal and two angles equal.
• A **scalene triangle** has all sides and angles of different measures.

The given triangle ABC is a scalene triangle because all side lengths, as given in the exercise, are different numbers: 37, 55, and 43.

Angle Calculation

Calculating the angles in a triangle when only the side lengths are known is a bit more complex. The Law of Cosines is very useful in such scenarios. It relates the lengths of the sides of a triangle to the cosine of one of its angles. This is especially helpful for scalene triangles where all sides are of different lengths.

The Law of Cosines states that for any triangle ABC:\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
To find one of the angles, such as angle A, rearrange the formula to solve for \(\cos(A)\) and compute the angle using inverse trigonometric functions:\[ \cos(A) = \frac{a^2 - b^2 - c^2}{-2bc} \]
Subsequently, calculate the angle using a calculator by evaluating the inverse of the cosine value obtained.

This approach allows the determination of precise angles in triangles when using accurate calculations with calculators, facilitating the entire process of angle determination in trigonometry.

Trigonometry

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It plays a vital role in various fields such as physics, engineering, and astronomy.

In trigonometry, several key concepts revolve around understanding angles and side lengths of triangles, utilizing functions like sine, cosine, and tangent. These functions help in the calculation of unknown parts of a triangle, given some initial known values. The **Law of Cosines** and **Law of Sines** are two primary formulas that are extensively used in solving problems related to triangles.

The Law of Cosines is particularly helpful in scenarios where you know:

• The lengths of all three sides of a triangle (SSS: Side-Side-Side situation).
• The lengths of two sides and the included angle in a triangle (SAS: Side-Angle-Side situation).

By applying these laws, trigonometry helps us solve problems involving triangles, which is essential for fields that require precise measurements and calculations.