Problem 8 - Algebra 2 and Trigonometry

Question

In \(3-14 :\) a. Determine the number of possible triangles for each set of given measures. b. Find the measures of the three angles of each possible triangle. Express approximate values to the nearest degree $$ a=9, c=10, \mathrm{m} \angle C=150 $$

Step-by-Step Solution

Answer

One triangle possible with angles: \(\angle A \approx 27^\circ\), \(\angle B \approx 3^\circ\), \(\angle C = 150^\circ\).

Step 1: Identify the given information

We have two sides of a triangle, \(a = 9\) and \(c = 10\), along with one angle, \(\angle C = 150^\circ\). This means that we are dealing with an angle-side-side (ASS) triangle configuration, which can sometimes produce 0, 1, or 2 possible triangles.

Step 2: Determine the possibility of a triangle using Law of Sines

Use the Law of Sines to find \(\sin A\): \(\frac{a}{\sin A} = \frac{c}{\sin C}\). Thus, \(\sin A = \frac{a \cdot \sin C}{c} = \frac{9 \cdot \sin 150^\circ}{10}\). Calculate \(\sin 150^\circ = 0.5\), so \(\sin A = \frac{9 \times 0.5}{10} = 0.45\). Since \(0 < \sin A < 1\), angle \(A\) exists, confirming one possible triangle.

Step 3: Calculate angle A

Using \(\sin A = 0.45\), find angle \(A\) with \(A = \sin^{-1}(0.45)\). Calculate to determine \(A \approx 27^\circ\).

Step 4: Determine angle B

Since triangles have interior angles summing to \(180^\circ\), calculate \(B = 180^\circ - 150^\circ - 27^\circ = 3^\circ\).

Step 5: Check for the second possible triangle

In ASS configurations, if angle A exists, a second triangle might exist with angle \(A' = 180^\circ - A = 153^\circ\), but \(A' + 150^\circ\) would exceed \(180^\circ\). Therefore, only one triangle is possible.

Key Concepts

Understanding the Angle-Side-Side (ASS) Configuration

The Angle-Side-Side (ASS) configuration is a unique setup in triangle problems. It involves two side lengths and a non-included angle. This can lead to multiple possible triangles due to the Law of Sines. The challenge with ASS is in determining whether any, one, or two triangles can be formed.

• Two sides of a triangle are known.
• An angle opposite one of those sides is known.

When solving an ASS configuration, the ambiguity arises from how the two sides and the angle relate to each other. Always start by trying to apply the Law of Sines to gain further insight into the configuration's potential solutions.

Exploring the Possible Number of Triangles

In an ASS configuration, it’s essential to determine the number of possible triangles that can be formed. Depending on the given values, there might be:

• No triangle, when the known angle and side do not satisfy the triangle inequality.
• One triangle, when a valid angle is found and additional angles do not violate possible angle sums.
• Two triangles, in cases where two different angles satisfy the conditions.

For this exercise, using the Law of Sines helped conclude that only one triangle is possible, since attempting to form a second corresponding angle leads to an impossible situation where angles would exceed 180 degrees.

Understanding the Interior Angles Sum of a Triangle

A fundamental property of triangles is that the sum of their interior angles is always 180 degrees. This property is a critical tool in analyzing triangle configurations. For any triangle with angles A, B, and C:\[A + B + C = 180^\circ\]When one or two angles are known, you can easily find the third by subtracting the known angles from 180 degrees. In our example, with given angles:

• Angle C = 150 degrees
• Angle A found to be approximately 27 degrees
• Therefore, B can be calculated as 180 - 150 - 27 = 3 degrees

Utilizing the Inverse Sine Function

The inverse sine function, denoted as \( \sin^{-1} \) or arcsin, is used to find an angle when its sine value is known. It plays an important role in resolving the ambiguity of the ASS configuration when using the Law of Sines to find unknown angles.When you have a sine value, like \( \sin A = 0.45 \), you use the inverse sine to find the angle A:\[A = \sin^{-1}(0.45)\]This calculation gives a specific value for angle A, approximately 27 degrees in this scenario. It is vital to check if a second angle possibility exists, by considering if another angle supplementary to the found angle could also maintain the configuration. Always verify to ensure that you don’t overlook additional possible solutions.