Question

question 6 of 10
suppose a triangle has two sides of length 3 and 4 and that the angle between these two sides is 60°. what is the length of the third side of the triangle?
a. ( 4 sqrt { 3 } )
b. ( sqrt { 3 } )
c. 3
d. ( sqrt { 13 } )

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(c^{2}=a^{2}+b^{2}-2ab\cos C\). Let \(a = 3\), \(b = 4\), and \(C=60^{\circ}\) (\(\cos60^{\circ}=\frac{1}{2}\)).
Substitute the values into the formula: \(c^{2}=3^{2}+4^{2}-2\times3\times4\times\frac{1}{2}\).

Step2: Calculate each term

First, calculate \(3^{2}=9\), \(4^{2}=16\), and \(2\times3\times4\times\frac{1}{2}=12\).
Then \(c^{2}=9 + 16-12\).

Step3: Simplify the expression

\(c^{2}=13\).

Step4: Solve for \(c\)

Take the square root of both sides: \(c=\sqrt{13}\).

Answer:

D. \(\sqrt{13}\)