Problem 26 - Algebra and Trigonometry

Question

In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. $$ a=30, b=40, A=20^{\circ} $$

Step-by-Step Solution

Answer

The given measurements can form one triangle. The triangle has angles A = 20 degrees, B = 26.4 degrees, C = 133.6 degrees and sides a = 30, b = 40, c = 88.5

Step 1: Check if a Triangle Can be Formed

Use the Law of Sines to check if a triangle can be formed using the provided measurements. The Law of Sines state that the ratio of a side of a triangle to the sine of its opposite angle is the same for each side. Calculate sin(B) = sin(A)*b/a. Plug in the values. Here, \(A=20^{\circ}\), \(a=30\) and \(b=40\). So, \(sin(B) = sin(20^{\circ})*40/30 = 0.433\). The sine of an angle in a triangle can only be between -1 and 1. Since 0.433 is in this range, a triangle can be formed.

Step 2: Determine the Possible Value for Second Angle

Use the arcsine function to calculate the possible values for angle B. The arcsine of a value gives the angle whose sine is that value. The arcsine of 0.433 is \(B=26.4^{\circ}\). But, in some SSA situations, two different triangles can be formed. For the second possible solution for angle B, subtract this value from 180: \(B' = 180 - 26.4 = 153.6^{\circ}\). However, the sum of angles in a triangle must be 180 degrees. With A=20 degrees, the angle B' is too large to form a triangle with angles A and B'. Therefore, only one triangle can be formed.

Step 3: Solve the Triangle

Now that the angle B has been found, solve for the remaining third angle C using the formula C = 180 - A - B. Substituting the values, we get \(C = 180 - 20 - 26.4 = 133.6^{\circ}\). Use the Law of Sines to solve for the remaining side c. From the formula c/sin(C) = a/sin(A), isolate c: \(c = sin(C)*a/sin(A)\). Substituting the values, we get \(c = sin(133.6^{\circ})*30/sin(20^{\circ}) = 88.5\).

Step 4: Provide the solution

The triangle has angles A = 20 degrees, B = 26.4 degrees, C = 133.6 degrees and sides a = 30, b = 40, c = 88.5

Key Concepts

SSA Triangle

An SSA triangle is a type of triangle where you know two sides and a non-included angle (the angle not between the two given sides). This situation can sometimes be tricky because SSA does not uniquely determine a triangle in all cases.

With SSA, you could end up with different results:

• One triangle
• Two possible triangles
• No triangle at all

This puzzle of SSA is famous in trigonometry as the "+ the ambiguous case." However, using the Law of Sines helps determine if and how many triangles can be formed with the given measurements. In our exercise, we started with the criteria of sides 30, 40, and an angle of 20 degrees. We needed to check if a triangle could even be formed first.

Arcsine Function

The arcsine function is an inverse trigonometric function used to find the angle whose sine value is known. In simpler terms, if you know the sine of an angle, the arcsine allows you to find the actual angle.

For the exercise, we calculated the sine of angle B using the ratio of the sides involved as per the Law of Sines. The sine value we ended up with was 0.433. Using the arcsine or \( ext{asin}\) function, we found that angle B is approximately 26.4 degrees.

A point to note when dealing with arcsine is that there are usually two potential angle solutions in the range [0, 180] degrees that can give the same sine value. This is why we also checked for another possibility, B' = 180 - 26.4, which didn't work for this specific problem.

Triangle Angle Sum

One of the most essential properties of a triangle is that the sum of its interior angles is always 180 degrees. This is true for any triangle, regardless of its type.

Once we determined that our angle B is 26.4 degrees, and we were given that angle A is 20 degrees, we could find the last angle (angle C) using this property.

By subtracting the sum of angles A and B from 180, we calculated:\[C = 180^\circ - 20^\circ - 26.4^\circ = 133.6^\circ\]
This calculation confirms that the triangle is valid and lets us confidently proceed to the next steps of solving the triangle fully.

Triangle Side Calculation

Once we know all angles of a triangle, the next step often involves finding unknown side lengths. In our case, we wanted to determine the length of side c.

The Law of Sines allows us to solve for side c using the proportion:\[\frac{c}{\sin(C)} = \frac{a}{\sin(A)}\]
By isolating c in this formula, we could solve for its length, substituting the known values:\[c = \frac{\sin(133.6^\circ) \times 30}{\sin(20^\circ)} \approx 88.5\]
This way, we have determined all the sides and angles for the triangle, completing our solution.