Problem 14 - Trigonometry

Question

If \(A=33^{\circ}, C=82^{\circ}\), and \(b=44 \mathrm{~cm}\), find \(B\) and then find \(c\).

Step-by-Step Solution

Answer

Angle \(B\) is 65°, and side \(c\) is approximately 48.45 cm.

Step 1: Understand Sum of Angles

A triangle's angles sum up to 180°. Given angles are \(A = 33^{\circ}\) and \(C = 82^{\circ}\). Find angle \(B\).

Step 2: Calculate Angle B

Subtract the sum of angle \(A\) and angle \(C\) from 180° to find \(B\). \(B = 180^{\circ} - (A + C) = 180^{\circ} - (33^{\circ} + 82^{\circ}) = 65^{\circ}\).

Step 3: Use the Law of Sines

We apply the Law of Sines to find side \(c\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). Use it to find \(c\).

Step 4: Set Up the Equation for c

From the Law of Sines, \(\frac{c}{\sin C} = \frac{b}{\sin B}\). Substitute known values: \(\frac{c}{\sin(82^{\circ})} = \frac{44}{\sin(65^{\circ})}\).

Step 5: Solve for c

Rearrange to find \(c\): \(c = \frac{44 \times \sin(82^{\circ})}{\sin(65^{\circ})}\). Calculate \(c\) using a calculator.

Key Concepts

Sum of Angles in a Triangle

In trigonometry, the sum of angles in any triangle is a fundamental concept. No matter the shape or size of the triangle, the interior angles will always add up to 180 degrees. This consistency allows us to deduce the measure of an unknown angle if two angles are known. In the given exercise, we already know angles A and C, which are 33° and 82°, respectively. By summarizing these, we see that the sum is 115°. Therefore, to find angle B, we subtract this sum from 180°:

• Step: 180° - (A + C) = 180° - 115° = 65°.

This results in angle B being 65°. It's this core understanding of the angles' sum that paves the path for solving other parts of the triangle, such as lengths of sides.

Law of Sines

The Law of Sines is a powerful tool used in solving triangles, particularly non-right triangles. This law relates the ratios of each side of a triangle to the sine of its corresponding opposite angle. It can be expressed as:

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

In our problem, we used the Law of Sines to find the unknown side, c. We already determined angle B and given side b, the relationship between b and B is:

• \( \frac{b}{\sin B} \)

By understanding that this ratio \( \frac{b}{\sin B} \) is equal to \( \frac{c}{\sin C} \), we set up the equation to solve for c. This approach showcases how laws like these provide interconnection between angles and sides in triangles. It’s particularly useful where information about non-right triangles is incomplete.

Solving Triangles

Solving triangles refers to finding all unknown angles and sides of a triangle when certain elements are provided. This could be a mixture of sides, angles, or even just one element of each. In the exercise, we started with known values, A, C, and side b, which allowed us to find B using the angle sum property first.

• Angle B = 65°

Next, we utilized the Law of Sines to find side c by establishing the relation between known and unknown values. This step required substitution and rearranging to isolate the variable c.

• Equation: \( c = \frac{44 \times \sin(82^{\circ})}{\sin(65^{\circ})} \)

Using a calculator computes the length of side c, highlighting the method of using trigonometric laws and properties for holistic triangle resolution. This approach not only solves the problem at hand but also builds a robust understanding to tackle similar tasks.