Problem 27 - Precalculus with Limits

Question

In Exercises 25-34, use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. \(A\ =\ 76^{\circ}\), \(a\ =\ 18\), \(b\ =\ 20\)

Step-by-Step Solution

Answer

So, the measure of angle B is 74.06 degrees, angle C is 29.94 degrees and side c is 9.98 units. There is only one solution for this triangle.

Step 1: Use the Law of Sines to Find Angle B

Using the Law of Sines, the measure of angle B can be found by using the formula \(sin B = \frac{b*sin A}{a}\).\nSo, \(sin B = \frac{20*sin(76^{\circ})}{18} = 0.961\).\nTo find angle B, calculate the arcsin of 0.961. This can be done utilizing a calculator having the arcsin function yielding \(B_1 = 74.06^{\circ}\). However, it can also be greater than 90 degrees, specifically \(B_2 = 180 - 74.06 = 105.94^{\circ}\). Therefore, there are two possible measures for angle B: \(B_1\ and\ B_2\).

Step 2: Check if Both Solutions are Possible

In a triangle, the sum of angles is 180 degrees. Therefore, it is important to check if angle C exists for both measures of angle B.\nFor \(B_1\), angle C can be calculated as \(C_1 = 180 - 76 - B_1 = 29.94^{\circ}\).\nFor \(B_2\), angle C can be calculated as \(C_2 = 180 - 76 - B_2 = -1.94^{\circ}\).\nSince negative angle is not possible, only the solution with \(B_1\) is possible.

Step 3: Use Law of Sines to Find Side c

Finally, to find side c for the only possible solution, apply the Law of Sines again, which gives \(c = a*\frac{sinC}{sinA}\).\nSo, \(c= 18*\frac{sin(29.94^{\circ})}{sin(76)} = 9.98\) round to decimal places.

Key Concepts

Triangle Solving

Solving triangles involves finding all the unknown sides and angles of a triangle. In most cases, you start with some known measurements, like an angle or a side, and from there, use mathematical rules to find the rest.
The type of information you have will guide which method to use.
For example:

• If you know two angles and one side (AAS or ASA), the Law of Sines is a great option.
• If you know two sides and the angle opposite one of them (SSA), you can use the Law of Sines, but be cautious of two possible solutions.
• If all you know are three sides (SSS) or two sides and the included angle (SAS), you might use the Law of Cosines, another powerful trigonometric rule.

Triangle solving is very much like solving a puzzle where recognizing the pattern often leads to the complete picture. It's a fundamental skill in trigonometry, essential for understanding the properties of shapes and their solutions.

Angle Calculation

Calculating angles is key to solving any triangle. You must be familiar with the properties of triangles, specifically the fact that the sum of angles in a triangle is always 180 degrees. This rule can help you find unknown angles once you have at least two known measurements.
For instance, in our scenario where

• Angle A is given as \(76^{\circ}\).
• Using the Law of Sines, you find angle \(B\). In this case, \(B_1 = 74.06^{\circ}\) could be a possible angle.
• To ensure the solution makes sense, check the third angle \(C\): \(C_1 = 180 - 76 - 74.06 = 29.94^{\circ}\).

Once you've calculated all angles correctly, you have determined the shape and size of the triangle accurately. Remember to verify your angles to make sure they don't exceed 180 degrees in total, ensuring the triangle remains possible.

Trigonometric Functions

Trigonometric functions allow us to solve triangles, bridges the gap between angles and sides, and are vital for calculations in trigonometry. In the context of triangle solving, these functions help establish relationships between triangle sides and angles.
The key trigonometric functions you'll likely encounter are sine, cosine, and tangent. Each of these functions can be calculated if you know a triangle's angle and its sides.
In this exercise, you used the sine function as part of the Law of Sines rule:

• The formula \(\sin B = \frac{b*\sin A}{a}\) gave the possible values for angle \(B\).
• You'll compute \(\sin B\), then apply the arcsine function to find \(B\), illustrating how we move back and forth from sides to angles.

Understanding and applying these trigonometric functions correctly ensures accurate sizing and solving of triangles across a wide range of applications.