Problem 45 - Trigonometry

Question

These questions are available for instructors to help assess if you have successfully met the learning objectives for this section. Use the law of sines to find \(C\) for triangle \(A B C\) if \(B=35^{\circ}, a=28\) feet, and \(b=19\) feet. a. \(23^{\circ}\) or \(87^{\circ}\) b. \(58^{\circ}\) c. \(58^{\circ}\) or \(87^{\circ}\) d. No triangle is possible

Step-by-Step Solution

Answer

Angle \( C \) is \( 87^{\circ} \). Option a (\( 23^{\circ} \) or \( 87^{\circ} \)) is correct.

Step 1: Recall the Law of Sines formula

The law of sines states that for any triangle \( ABC \), the ratio of a side length to the sine of its opposite angle is equal: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]. We will use the known sides and angle to find angle \( C \).

Step 2: Set up the equation for \( A \)

Using the law of sines, we want to find angle \( A \). From our given values, we have \( \frac{28}{\sin(A)} = \frac{19}{\sin(35^{\circ})} \).

Step 3: Solve for \( \sin(A) \)

Rearrange the equation to solve for \( \sin(A) \). Calculate \( \sin(A) = \frac{28 \times \sin(35^{\circ})}{19} \). Compute \( \sin(35^{\circ}) \) and multiply it by 28, then divide by 19.

Step 4: Calculate \( \sin(35^{\circ}) \)

Using a calculator, compute \( \sin(35^{\circ}) \approx 0.5736 \).

Step 5: Evaluate \( \sin(A) \)

Substitute the computed sine value: \( \sin(A) \approx \frac{28 \times 0.5736}{19} \approx 0.8457 \).

Step 6: Determine angle \( A \)

Since \( \sin(A) = 0.8457 \), find \( A = \sin^{-1}(0.8457) \) using the inverse sine function. Angle \( A \approx 57.98^{\circ} \).

Step 7: Verify triangle angle sum

The sum of angles in a triangle is 180°. Therefore, \( C = 180^{\circ} - B - A \). Calculate \( C = 180^{\circ} - 35^{\circ} - 57.98^{\circ} = 87.02^{\circ} \approx 87^{\circ} \).

Step 8: Double-check for alternate solutions

Given the sine rule, there could be two possible angles for \( A \). However, \( A \) and \( B \) are already less than 90°, and adding any larger value will violate the triangle inequality. Hence, \( C = 87^{\circ} \) is valid.

Key Concepts

Triangle Trigonometry

Understanding triangle trigonometry is crucial when it comes to solving problems about triangles, especially in the context of angle calculations like in the given exercise. Triangles in trigonometry typically involve methods such as the Law of Sines and the Law of Cosines which allow us to find unknown angles or sides.

The Law of Sines is a powerful tool in triangle trigonometry. It relates the sides and angles of a triangle using the formula:

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

This formula indicates that the ratio of the length of a side of the triangle to the sine of its opposite angle is constant for all three sides and angles.

Understanding this principle enables you to solve for unknown angles in a triangle when certain side lengths and an angle are known, as demonstrated in the exercise by determining the missing angle \(C\) using the provided angles and lengths.

Angle Calculation

Angle calculation in triangles involves using known angles and side lengths to determine unknown values. This is often seen in problems like the given exercise, where you are tasked with finding a specific angle using given data.

Step 1 is to employ the Law of Sines as described earlier. In our example:

• Find \(A\) using: \( \frac{28}{\sin(A)} = \frac{19}{\sin(35^{\circ})} \)
• Compute \( \sin(35^{\circ}) \) using a calculator: approximately \( 0.5736 \)
• Solve for \( \sin(A) \) by rearranging and substituting the known values.

Once \( \sin(A) \) is computed, use the inverse sine function, denoted as \( \sin^{-1} \), to find the angle \(A\). This allows students to understand each calculation step-by-step, emphasizing mastery over each part of the process.

Ultimately, angle calculation hinges on being methodical and verifying computations through each step, ensuring accuracy like in finding angle \(C\) (\(180^{\circ} - B - A\)).

Triangle Inequality

The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This is a fundamental rule that governs the possibility of forming triangles.

In the context of angle calculation problems, the triangle inequality acts as a verifier of feasible results. For example, after calculating angles \(A\) and \(B\), angle \(C\) is determined by ensuring the total angle sum equals \(180^{\circ}\), which must respect the individual inequalities of internal angles being less than \(90^{\circ}\).

In the original problem setup, checking if the sum of \(A \) and \(B \) is less than \(180^{\circ} \) confirms that a valid triangle can still be formed. Moreover, validating that no combination of side lengths and angles permits a sum or size to exceed logical bounds (exceeding \(180^{\circ}\) in angles, for example) further supports confidence in the calculated results. As in our solution, checking the case of possible alternate solutions for \(A\) and ensuring compliance with these inequalities confirms the angle \(C = 87^{\circ}\).