Problem 34 - Precalculus with Limits

Question

Solving a Triangle, determine whether the Law of Sines or the Law of Cosines is needed to solve the triangle. Then solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$ a=11, \quad b=13, \quad c=7 $$

Step-by-Step Solution

Answer

Calculate angle A using the Law of Cosines and round it to two decimal places. Repeat the same procedure for angle B. Angle C is calculated as \( C = 180 - A - B \). If angle C is between 0° and 180°, the triangle has one solution: angles A, B, and C. Otherwise, the triangle has no solution or could not exist.

Step 1: Find the first angle using the Law of Cosines

We can use the formula \[ A = \cos^{-1} \left( \frac{{b^2 + c^2 - a^2}}{{2bc}} \right) \]Plugging the numbers in, we get\[ A = \cos^{-1} \left( \frac{{13^2 + 7^2 - 11^2}}{{2*13*7}} \right) \]

Step 2: Calculate A

Calculate the expression to find the value of angle A. Remember to round your answer to two decimal places.

Step 3: Find the second angle using the Law of Cosines

Repeating the first two steps, you find the second angle B using the formula \[ B = \cos^{-1} \left(\frac{{a^2 + c^2 - b^2}}{{2ac}} \right) \]which gives \[ B = \cos^{-1} \left(\frac{{11^2 + 7^2 - 13^2}}{{2*11*7}} \right) \]

Step 4: Calculate B

Calculate the expression to find the value of angle B. Remember to round your answer to two decimal places.

Step 5: Find the third angle using the sum of angles in a triangle

The sum of angles in a triangle is 180 degrees, so \( C = 180 - A - B \)

Step 6: Calculate C

Calculate the expression to find the value of angle C. If the value of C is between 0° and 180°, there is one solution, otherwise, if its value is 0° or 180°, there are no solutions, and if it is negative, the inputs could not form a triangle.

Step 7: Display the Solution

The values of the angles A, B, and C rounded to two decimal places are the final solution for this exercise.

Key Concepts

Law of Sines

The Law of Sines is a fundamental concept in trigonometry used to find unknown parts of a triangle when certain conditions are met. This law relates the lengths of a triangle's sides with the sines of its opposite angles. It is particularly useful in situations where two angles and a side (AAS or ASA) are known, or two sides and one angle (SSA) where one of the sides is opposite the known angle. The Law of Sines formula is: \[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \]

• a, b, c: The lengths of the sides of the triangle.
• A, B, C: The angles opposite these respective sides.

When solving triangles, you would typically employ the Law of Sines to find missing angles or sides, given sufficient initial information. However, when dealing with certain ambiguities, such as the ambiguous case (SSA), the Law of Sines might suggest two possible solutions for the triangle. In our original exercise, using the Law of Sines would be more fitting if the given parts of the triangle allowed for direct matching with this proportion method. If angles were provided, it could simplify problems significantly.

Law of Cosines

The Law of Cosines is another essential trigonometric tool, especially useful in scenarios where you know two sides and the included angle (SAS) or all three sides (SSS), but no angles. The Law of Cosines helps you find unknown angles or sides in any triangle, not restricted to right triangles.The Law of Cosines is expressed by the following formula: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] This formula can also be rearranged to solve for angles:\[ C = \cos^{-1} \left( \frac{a^2 + b^2 - c^2}{2ab} \right) \]In the original exercise, we used this law to calculate the angles A and B based on the side lengths provided. The choice of the Law of Cosines was facilitated by the triangle type (three sides given), and it allowed direct calculation of an angle once two sides and their interposed angle were known. The law is particularly robust as it works regardless of whether the triangle is obtuse, acute, or right, providing reliable angle information even when all sides are known.

Angle Calculation

Calculating angles in a triangle involves several steps and can utilize several methods, depending on the known elements. In our solution, we calculated the angles using a step-by-step approach with the help of trigonometric laws. After determining the initial angles using the Law of Cosines, the final angle can be quickly found because the sum of angles in any triangle is always 180 degrees. If angles A and B are known, angle C is given by:\[ C = 180^{\circ} - A - B \]This property allows for completion of a triangle's angle list once two are determined. This calculation helps assure that the triangle is valid; any result that does not adhere to a total of 180 degrees may indicate an error in calculation or that the inputs do not form a triangle. For our problem, knowing these calculations ensures that we round our angles appropriately to two decimal places as a final step, meeting precision requirements needed for more complex applications or evaluations.

Triangle Geometry

The basics of triangle geometry are fundamental to understanding how to solve any triangle problem. Triangles have three sides, three angles, and several distinct properties that determine how they are categorized, such as being scalene, isosceles, or equilateral. In solving triangle problems, key properties you need to account for are:

• All angles within a triangle sum to 180 degrees.
• The longest side is always opposite the largest angle.
• Two sides must sum to more than the third side to form a valid triangle.

Using knowledge of these properties, you can double-check your solution for consistency and validity. In our exercise, triangle geometry principles guide us in ensuring that calculated angles and side relationships confirm the constructed triangle is possible and accurate. Each calculation made should fit within these criteria. Checking against them reinforces not only correct usage of laws but also helps to avoid simple calculation errors in the process.