use cosine law, which says that c² = a²+b²-2ab cosC.
in the cosine law formula, angle C is the angle opposite to side c. so a and b should be opposite to angles that you do NOT know. that is when you can know to use the cosine law
here, a = 12 yd, b = 17yd, and C = 53°. so we can find QS using cosine law. if c = QS then
c² = 12²+17²-2(12)(17) cos53
c = √[ 12²+17²-2(12)(17) cos53 ]
c = 13.69158 yd = QS
them you can use sine law to find the angles
for angle S
sin S / 12 = sin 53 / 13.69158
sin S = 12 sin 53 / 13.69158
S = sin⁻¹(12 sin 53 / 13.69158)
S = 44.42417°
for angle Q
sin Q / 17 = sin 53 / 13.69158
Q = sin⁻¹(17 sin 53 / 13.69158)
Q = 82.5758
your answers to nearest tenth:
m∠Q = 82.6°
m∠S = 44.4°
QS = 13.7 yd
for question 14, we establish that the angle of depression is in that spot there that forms with the dashed lines. then we can say that angle is the same as that angle in the triangle that forms with the ladder. then, since that's a right angle, we use cosine to find the length between the bottom of ladder to the wall
cos 62 = d / 24
d = 24*cos62
d =11.3 ft