Problem 5 - Precalculus : Building Concepts and Connections

Question

Use the Law of Cosines to find the umknoten side of each of the given triangles. The standard notation for labeling of triangles is used. Round anstoers to four decimal places. $$b=8, c=4, A=75^{\circ}$$

Step-by-Step Solution

Answer

The length of side a, calculated using the Law of Cosines and rounded to four decimal places, is the final answer.

Step 1: Understand and Write Down the Law of Cosines Formula

The Law of Cosines states that there is a relationship between the sides of a triangle and the cosine of one of its angles. For a triangle with sides of lengths a, b, c and an angle A opposite side a, the formula can be expressed as follows: \( a^{2} = b^{2} + c^{2} - 2bc \cos(A) \)

Step 2: Substitute Known Values into Equation

Now it's time to substitute the values from the problem into the formula. From the problem, we have \( b = 8 \), \( c = 4 \), and \( A = 75^\circ \). Remember to convert the angle from degrees to radians before calculating cosine, as most calculators use radian mode. \( A_{rad} = \frac{75\pi}{180} \). Now the Law of Cosines equation for this problem is: \( a^{2} = 8^{2} + 4^{2} - 2(8)(4) \cos \left( \frac{75\pi}{180} \right)\)

Step 3: Calculate the Unknown Side

Perform the arithmetic operations to calculate the value of \(a^{2}\). After finding \(a^{2}\), take the square root of both sides of the equation to find the length of side a. We round this final answer to four decimal places.