Problem 28 - Precalculus with Limits

Question

Using the Law of Sines. Use the Law of Sines to solve (if possible) the triangle. If two solutions exist, find both. Round your answers to two decimal places. $$A=76^{\circ}, \quad a=34, \quad b=21$$

Step-by-Step Solution

Answer

This could lead to two potential sets of solutions for the triangle's dimensions, considering the values of the angles and lengths of sides calculated through steps. It's important to take into account the limitations of angles in a triangle (they must be less than 180 degrees) and the validity of the side lengths (they must be positive). Round all the answers to two decimal places.

Step 1: Use the Law of Sines to find another angle

First, use the known angle \(A\) and side \(a\) along with the given side \(b\) to solve for angle \(B\) using the Law of Sines. This is done as follows: \(\frac{a}{sinA} = \frac{b}{sinB}\), rearrange the equation for \(sinB\): \(sinB = \frac{b \cdot sinA}{a}\). Substitute the given values into the equation to find \(sinB\).

Step 2: Calculate the value of angle B

Use the calculated value of \(sinB\) to find the value of angle \(B\) using the inverse sine function, also take into consideration that the sine function is positive in both the first and second quadrants, hence two angles could be possible. \[ B_1 = sin^{-1}(sinB) \] \[ B_2 = 180 - B_1 \] Note: It's important to ensure that the sum of the calculated angles does not exceed 180 degrees.

Step 3: Calculate the third angle

To find the third angle \(C\), subtract the sum of angles \(A\) and \(B\) from 180. This is because the internal angles of a triangle always add up to 180 degrees. This is done as follows: \[ C = 180 - A - B \] Calculate the third angle for both cases.

Step 4: Use the Law of Sines to find the third side

Finally, use the Law of Sines with angle \(C\) to find side \(c\), using the formula: \( c = a \cdot \frac{sinC}{sinA} \) and the previously calculated value for \(sinC\). Calculate this for both situations of angle C.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that delves into the relationships between the sides and angles of triangles. The Greek word