Problem 22 - Algebra 2 and Trigonometry

Question

Aaron wants to draw \(\triangle A B C\) with \(A B=15\) inches, \(B C=8\) inches, and an area of 40 square inches. a. What must be the sine of \(\angle B ?\) b. Find, to the nearest tenth of a degree, the measure of \(\angle B .\) c. Is it possible for Aaron to draw two triangles that are not congruent to each other that satisfy the given conditions? Explain.

Step-by-Step Solution

Answer

a. \(\sin(\angle B) = \frac{2}{3}\). b. \(\angle B \approx 41.8^\circ\). c. Yes, two triangles can be formed with the given conditions.

Step 1: Understand the Problem

We have a triangle with given sides and area. We need to use this information to find \(\sin(\angle B)\), which will then help determine the measure of \(\angle B\). Additionally, we'll determine if two non-congruent triangles can satisfy the same conditions.

Step 2: Use the Area Formula

The area of a triangle can be expressed as \(\text{Area} = \frac{1}{2}ab\sin(C)\). For \(\triangle ABC\), we have \(a = 15\), \(b = 8\), and \(\text{Area} = 40\). Plugging these into the area formula gives:\[40 = \frac{1}{2} \times 15 \times 8 \times \sin(\angle B)\]

Step 3: Solve for \(\sin(\angle B)\)

Rearrange the equation from Step 2 to solve for \(\sin(\angle B)\):\[40 = 60\sin(\angle B) \\sin(\angle B) = \frac{40}{60} = \frac{2}{3}\]So, the sine of angle \(B\) is \(\frac{2}{3}\).

Step 4: Determine \(\angle B\)

Use the inverse sine function to find \(\angle B\):\[\angle B = \sin^{-1}\left(\frac{2}{3}\right) \]Using a calculator, find \(\angle B \approx 41.8^\circ\). To the nearest tenth of a degree, \(\angle B\) is approximately \(41.8^\circ\).

Step 5: Assess Possibility of Two Non-Congruent Triangles

For two triangles to be non-congruent, they must differ in at least one angle or side. In this problem, the presence of a second triangle is determined by the existence of an alternative angle for \(\angle B\) that satisfies all given conditions. Since \(\sin(\angle)\) has an ambiguous case (angle supplementing each other), calculate \(\angle B\) for its supplement:\[180^\circ - 41.8^\circ \approx 138.2^\circ\]Since \(\angle B\) can also be approximately \(138.2^\circ\), it's possible to form another triangle under these conditions.

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry that relates the ratio of the length of the opposite side to the hypotenuse of a right-angled triangle. It is denoted as \( \sin(\theta) \), where \( \theta \) represents an angle. In our exercise, we use the sine function to explore relationships within triangle \( \triangle ABC \).
The sine of an angle \( \theta \) can be expressed as:

• \( \sin(\theta) = \text{Opposite} / \text{Hypotenuse} \)

In acute or obtuse triangles, when sides and one angle are given, the sine function helps solve for other angles or sides using sine rules. For \( \triangle ABC \), finding the sine of angle \( B \) helps calculate the angle using given sides and area.
In trigonometric calculations involving areas of triangles, the sine function becomes especially important, as seen in the formula for the area of a triangle \( \text{Area} = \frac{1}{2}ab\sin(C) \), where \( a \) and \( b \) are the lengths of two sides and \( \sin(C) \) is the sine of the angle between them.

Area of a Triangle

The area of any triangle is essential in understanding its size and properties. The formula connecting the area of a triangle with its side lengths and the sine of one of its angles is

• \( \text{Area} = \frac{1}{2} ab \sin(C) \)

where \( a \) and \( b \) are the sides enclosing the angle \( C \). This formula is incredibly useful when two sides and the included angle are given, as it can provide the area thoroughly.
In our example, given that the area is 40 square inches, side lengths 15 and 8 inches, we insert these values into the formula to find \( \sin(\angle B) \). This approach allows us to rearrange the area formula and solve

• \( 40 = \frac{1}{2} \times 15 \times 8 \times \sin(\angle B) \)

Finding the sine of the angle is thus a direct path to understanding the specific aspects of the triangle.

Angle Measurement

Understanding angle measurement is critical in trigonometry. It often involves using trigonometric ratios to determine the size of an angle in degrees or radians. In our exercise, after determining \( \sin(\angle B) = \frac{2}{3} \), we then find the angle measurement using the inverse sine function:

• \( \angle B = \sin^{-1}\left(\frac{2}{3}\right) \)

This operation often requires calculators for precision and returns an angle in degrees. Calculators or mathematical software are frequently used to find approximate angle measures efficiently.
The angle measurement of \( \angle B \) approximates to \( 41.8^\circ \) after using the inverse function. Knowing how to measure angles accurately is crucial, particularly when these measurements will influence the possibility of forming other triangles with given sides.

Ambiguous Case of Triangles

The ambiguous case arises specifically in situations involving the Law of Sines, typically when two sides and a non-included angle are known. This situation often leads to two different valid triangles, which is the case when calculating \( \angle B \) in \( \triangle ABC \).
With \( \sin(\angle B) \) calculated, the ambiguous case considers both \( \angle B \) and its supplement:

• \( 41.8^\circ \)
• \( 138.2^\circ = 180^\circ - 41.8^\circ \)

Both angles provide valid triangles, leading to potential non-congruent outcomes that follow the initial conditions. This is called the ambiguous case because when \( \sin(A) = c \), there may be two different angles that satisfy the equation, underlining the importance of context in interpreting results within triangle geometry.