Problem 24 - Precalculus : Building Concepts and Connections

Question

If at least one triangle with the given characteristics exists, solve the triangle; otherwise, state that there is no solution. If there are treo solutions, give both of them. The standard notation for labeling of triangles is used. $$A=25^{\circ}, a=7, b=9$$

Step-by-Step Solution

Answer

The first triangle has sides of lengths approximately 7, 9, and 11.45 and angles of approximately 25, 36.58, and 118.42 degrees. The second triangle does not exist as the calculated angle for C' is a negative value.

Step 1: Using Law of Sines

First will be applying the law of sines to find angle B. The law of sines can be written as \(\frac{a}{\sin A} = \frac{b}{\sin B}\). We can solve for B which gives us \(B = \sin^{-1} (\frac{b \sin A}{a}) = \sin^{-1} (\frac{9 \sin 25}{7})\).

Step 2: Calculate first possible value for B

We plug the known values into the equation from the first step. Calculating this gives us that the first possible value for B is roughly 36.58 degrees.

Step 3: Calculate second possible value for B

Since sine is positive in both the first and second quadrants, there can potentially be a second solution for B. This can be found using the formula \(B' = 180 - B = 180 - 36.58\). Computing this gives our second potential value for angle B, roughly 143.42 degrees.

Step 4: Test for Valid Triangle

In order for a triangle to exist, the sum of its angles must be equal to 180 degrees. Now one can test both angle possibilities. For the first possible value of B, we can find angle C by using the formula \(C = 180 - A - B = 180 - 25 - 36.58\), and for the second possible value of B, we apply the same formula but with B', \(C' = 180 - A - B' = 180 - 25 - 143.42\). Both C and C' should be positive values if these triangles exist.

Step 5: Solve for Side C

If viable triangle solutions are present, solve for the missing side length using the law of sines again. The formula for side c will be \(c = \frac{a \sin C}{\sin A}\). Repeat for the second potential scenario to find either c or c'.

Key Concepts

Solving Triangles

Understanding how to solve triangles involves finding the unknown angles and sides when you're given certain initial measurements. Common scenarios include being given side-angle-side (SAS), angle-side-angle (ASA), or side-side-side (SSS) combinations. For the given exercise, the problem presents an angle-side-side (ASS) case, where you have one angle and two sides. Utilizing the Law of Sines allows for the determination of unknown angles and, subsequently, any remaining sides.

When solving triangles, it's crucial to be aware of the possibility of multiple solutions. In particular, the ASS scenario can result in one, two, or no solutions due to the ambiguous case of the Law of Sines. This is because for certain angles and lengths, there could be two different triangles that satisfy the given conditions, as illustrated in step 2 and step 3 of the problem solution. It is only by considering the triangle angle sum rule and checking the validity of potential solutions that we can ascertain the correct scenario.

Trigonometric Functions

Trigonometric functions are fundamental tools in solving triangles, particularly in the realm of non-right triangles. The three primary trigonometric functions are sine (sin), cosine (cos), and tangent (tan), each relating the angles of a triangle to the ratios of its sides. The sine function, used prominently in the Law of Sines, relates the length of the side opposite an angle to the length of the triangle's longest side, the hypotenuse, in right triangles, and to other sides using the Law of Sines for oblique triangles.

In our exercise, the function \( \sin^{-1} \) or arc-sine is used, which serves to find the angle when the ratio of the opposite side to the hypotenuse is known. An important aspect of trigonometric functions to remember is that they can have the same positive value for two different angles within a 0° to 180° range — one in the first quadrant and one in the second. This duality leads to the potential two solutions encountered when applying the Law of Sines.

Triangle Angle Sum

The triangle angle sum is a theorem that states the sum of the interior angles of a triangle is always equal to 180 degrees. This fact is integral to solving triangle problems, as it provides a method to calculate missing angles when at least two angles are known. Throughout the steps of the given solution, this rule is repeatedly employed to confirm the validity of each potential triangle by ensuring that the sum of the calculated angles does not exceed 180 degrees.

For instance, after finding one or two possible values for angle B using the Law of Sines, you must then check if these angles lead to a valid angle C which, together with the given angle A and the calculated angle B (or B'), adds up to 180 degrees. Should this not be the case, it would signify the absence of a viable solution. This principle is beautifully illustrated in step 4 of the solution, showing how both potential angle B values lead to a positive angle C, and thus, two valid triangles.