# Trig-sine, cosine law

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Two fire stations,P and Q,are 20 km apart. A ranger at station Q sees a fire 15.0 km away. If the angle between the line PQ and the line from P to the fire is 25degrees, how far to the nearest tenth of a kilometre, is station P from the fire?

• Trig-sine, cosine law -

Now this one is easy to see and draw.

direct application of cosine law,

x^2 = 20^2 + 15^2 - 2(20)(15)cos 25°
= 81.21533
x = 9.01 km

• Trig-sine, cosine law -

Um the answer I was given was 5.7km or 30.5km
So how wud u get these answers?

• Trig-sine, cosine law -

Yup, I made a mistake, I didn't mind my P's and Q's
I now have the fire at F, and PQ=20, FQ=15 and angle QPF=25°

so we have the ambiguous case of the sine law.
let PF = x
first we have to find angle F
sinF/20 = sin25/15
sinF = .5635
angle F = 34.3° , then angle Q = 120.7
or angle F = 145.7°, then angle Q = 9.3°

case 1:
x/sin 9.3 = 15/sin25
x = 5.73

case 2
x/sin120.7 = 15/sin25
x = 30.52
x =

• Trig-sine, cosine law -

Thank you so much:)

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