Problem 73 - Algebra and Trigonometry

Question

In Exercises 71–74, determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving an SSA triangle using the Law of sines, my calculator gave me both the acute and obtuse angles \(B\) for which \(\sin B=0.5833\)

Step-by-Step Solution

Answer

The statement makes sense. For a given value of \(\sin B=0.5833\), there could be two possible values for the angle B - one acute and one obtuse. This is because of the periodic nature of the sine function.

Step 1: Understanding of Sin Values

Firstly, let's understand some basics about sine (sin) function. Sine function gives the ratio of the length of the side of the triangle opposite to the angle to the length of the longest side of the triangle (hypotenuse). It's value ranges from -1 and 1. For acute angles (less than 90 degrees), sin values are positive and less than 1, and for obtuse angles (greater than 90 degrees), sin values are still positive but will be reverse to its corresponding acute angle as the sine function is periodic.

Step 2: Understanding Scenario in the Problem

As per the given problem, it looks like by using the law of sines to solve an SSA triangle, we find that \(\sin B=0.5833\). Since the given \(\sin B\) value is positive (and less than 1), it could correspond to two different cases: an acute angle (in the first quadrant) as well as an obtuse angle (in the second quadrant). This is possible due to the cyclic nature of the sine function, where sin(B) = sin(180-B), where B is an acute angle.

Step 3: Conclusion of Analysis

The statement does make sense. For a given value of \(\sin B\), there could be two possible values for angle B - one acute and one obtuse, namely B and 180-B (where B<90). This is a result of the way the sine function works in the unit circle. However, whether both angles are feasible results for a triangle's internal angle depends on the specifics of the triangle.

Key Concepts

SSA triangle

The SSA triangle configuration, also known as Side-Side-Angle, arises when two sides and a non-included angle are known. This situation is particularly interesting because it can lead to multiple triangle solutions or even no solution at all. SSA is unique in its ambiguity since, without careful consideration, the known angle might not be directly opposite one of the known sides.
In such cases, you could end up having:

• One valid solution.
• Two valid solutions.
• No solution.

To determine which scenario applies, we often rely on the Law of Sines, examining the possible values for a given angle to figure out the valid triangle configurations.

sine function

The sine function is a fundamental mathematical concept used in trigonometry to relate the angle of a triangle to the ratio of the length of the opposite side and the hypotenuse. The function varies in a predictable manner, repeating every 360 degrees. Within the range of an angle from 0 to 360 degrees, the sine function completes its cycle once, creating different values depending on the angle's position.
Key features of the sine function include:

• The sine of an angle is positive in both the first (acute angles) and second quadrants (obtuse angles).
• The function is periodic, which means \( \sin(\theta) = \sin(180^\circ - \theta)\) for any angle \(\theta\).
• Its range is limited to -1 and 1, making it a bounded function.

Being familiar with these properties helps solve triangles, especially in ambiguous cases like those in SSA configurations.

acute and obtuse angles

Understanding acute and obtuse angles is crucial when working with trigonometric problems, especially in triangles. An acute angle is any angle less than 90 degrees, while an obtuse angle is greater than 90 degrees but less than 180 degrees. These definitions are simple yet play a crucial role in triangle properties and the cyclic nature of the sine function.
Several important points about these angles include:

• Acute angles are always found in the first quadrant of the unit circle where all trigonometric functions, including sine, are positive.
• Obtuse angles are found in the second quadrant, where the sine function remains positive, which is why a single sine value can correspond to both an acute and an obtuse angle (i.e., \(\sin(B) = \sin(180^\circ - B)\)).

When solving triangles, recognizing how acute and obtuse angles relate to the sine function helps in accurately identifying possible solutions.

triangle properties

Triangles are fundamental geometric shapes characterized by three sides and three internal angles. Understanding their properties is essential when solving problems related to them. One property critical to remember is that the sum of the internal angles of any triangle is always 180 degrees. This constraint guides solving many triangle problems, including when ensuring the feasibility of angle solutions in the SSA scenario.
Key properties include:

• Each internal angle must be between 0 and 180 degrees.
• For SSA triangles with two angles provided by the sine function, it's important to check if the set configuration satisfies the triangle angle sum property.
• A triangle is valid only if the angles add up to exactly 180 degrees, and each angle is positive.

These insights about triangle properties assure that any determined angles align with feasible triangles.