Question

find the length of side c.

b

a = 47 c = ?

c 73° a

b = 45

law of cosines: c² = a² + b² - 2ab cos(c)

Explanation:

Step1: Recall Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\). Here, \(a = 47\), \(b = 45\), and \(C=73^{\circ}\).

Step2: Substitute values into formula

First, calculate \(a^{2}=47^{2}=2209\), \(b^{2}=45^{2} = 2025\), and \(2ab\cos(C)=2\times47\times45\times\cos(73^{\circ})\). Calculate \(\cos(73^{\circ})\approx0.2924\). Then \(2\times47\times45\times0.2924=2\times47\times45\times0.2924 = 47\times90\times0.2924=4230\times0.2924\approx1236.852\).
Now, \(c^{2}=2209 + 2025- 1236.852=4234 - 1236.852 = 2997.148\).

Step3: Take square root

Take the square root of \(2997.148\) to find \(c\). \(c=\sqrt{2997.148}\approx54.75\).

Answer:

\(\approx54.75\) (or more precise value depending on calculation of \(\cos(73^{\circ})\))