Question

Please help with this problem see image

Answer 1

Since we have 3 sides but no angles, or known as a SSS problem, we cannot use the Law of Sines, which requires 1 angle.
Therefore we must resort to the Law of Cosines, which states:
${c}^{2} = {a}^{2} + {b}^{2} - (2ab \cos(C)$
let's make <F = C, therefore c is opposite C and = 5 yd we'll make a = 6 yd and b = 7 yd
So now let's plug in and solve for Cos(C):
${c}^{2} = {a}^{2} + {b}^{2} - (2ab \cos(C)) \\ - 2ab \cos(C) = {c}^{2} - {a}^{2} - {b}^{2} \\ - 2ab \cos(C) \div - 2ab \\ = ({c}^{2} - {a}^{2} - {b}^{2}) \div - 2ab$
$\cos(C) = \frac{{c}^{2} - {a}^{2} - {b}^{2}}{- 2ab} = \frac{{5}^{2} - {6}^{2} - {7}^{2}}{- 2(6)(7)} \\ = \frac{25 - 36 - 49}{- 2(42)} = = \frac{{5}^{2} - {6}^{2} - {7}^{2}}{- 84}$
$\cos(C)= \frac{ - 60}{ - 84} = \frac{5}{7} \\ C = { \cos(\frac{5}{7} ) }^{ - 1} = 44.42 \: degrees$
Therefore the answer is A) 44° !!

Answer 2

Using Cosine Rule:

5² = 7² + 6² - 2(7)(6) cos(F)

25 = 85 - 84cos(F)

84 cos(F) = 60

cos(F) = 5/7

F = cos⁻¹(5/7) 

F = 44°

Answer:  44°