Problem 22 - 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. $$ a=6.1, b=4, A=162^{\circ} $$

Step-by-Step Solution

Answer

The given measurements can form one triangle. The measures are: side a = 6.1, side b = 4, side c = 0.2; angle A = 162°, angle B = 16.3°, angle C = 1.7°.

Step 1: Determine if a triangle could exist

To determine if a triangle could exist, one can use the sine rule. The sine rule states, \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Leaving us with an expression for B: \(B = \arcsin\left(\frac{b\cdot\sin A}{a}\right)\). \n Putting given values will give B = 16.3°.

Step 2: Check if there is an ambiguous case

Ambiguous case occurs if \(a < b \sin A\) and \(a \neq b\). If \(\sin B = \frac{b \sin A}{a}\) is more significant than 1, a triangle does not exist. However, \(\sin B = 0.6556 < 1\), thus a triangle can exist.

Step 3: Find angle C for the one triangle solution

The sum of the three angles in a triangle should be 180°. Using this theorem, we can find the other angle: \(C = 180 - A - B = 180 - 162 - 16.3 = 1.7°.\)

Step 4: Find the size of side c

To find the length of the missing side, we can use the law of sines again: \(\frac{c}{\sin C} = \frac{a}{\sin A}\). Solving for c gives \(c = \frac{a \sin C}{\sin A} = \frac{6.1 \sin 1.7°}{\sin 162°} = 0.2 \: (\text{rounded to nearest tenth}).\)

Key Concepts

Law of Sines

Understanding the Law of Sines is essential in the world of trigonometry, especially when dealing with triangles. It's a formula that relates the lengths of the sides of a triangle to the sines of its angles. Specifically, for any triangle with sides a, b, and c, and respective opposite angles A, B, and C, the Law of Sines can be written as:
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
To put this into context, let's say we have two sides and one non-included angle (SSA), as in the problem at hand. We can start by solving for one of the unknown angles using the Law of Sines. Since we know side 'a' and its opposite angle 'A', and we have another side 'b', we can rearrange the equation to solve for angle 'B':
\[ B = \arcsin\left(\frac{b \cdot \sin A}{a}\right) \]
Using the given values, we find that angle B is approximately 16.3°. This step is crucial as it kicks off the process of solving triangles using the information available.

Ambiguous Case

The Ambiguous Case, also known as SSA Ambiguity, arises in trigonometry when dealing with triangles where two sides and a non-included angle are given. This situation can lead to zero, one, or two possible triangles.

To decode this ambiguity, certain conditions must be examined. First, for a real triangle to exist, the sine of the calculated angle must be less than or equal to one, since the sine function's range is within -1 and 1. If the calculated sine value exceeds this range, no triangle exists.

In our example, since \[ \sin B = \frac{b \sin A}{a} \] yields a result less than 1, it confirms the possibility of a triangle. However, the true test of ambiguity checks whether side 'a' is shorter than or equal to side 'b' times the sine of angle 'A'. If so, and side 'a' is not equal to side 'b', two triangles can be formed. If not, it results in a unique triangle. Our calculation didn't meet this criterion, indicating a single solution.

Solving Triangles

Solving triangles requires determining the remaining unknown lengths and angles when certain dimensions and angles are provided. For any triangle, the sum of the interior angles should always equal 180 degrees. Thus, after computing two angles, the third can easily be found.

In the given problem, after finding angle 'B', we can use the simple equation:
\[ C = 180^\circ - A - B \]
to find the third angle, 'C'. This step provides us with the full set of angles for the triangle. The final side 'c' remains to be determined.
\[ c = \frac{a \cdot \sin C}{\sin A} \]
Conveniently, the Law of Sines comes to our rescue again, providing the length of the missing side based on our known angle 'C' and the initial side 'a'. Through this sequence of steps, we solve the triangle fully, respecting the given measurements and rounding rules.