How find third side with 4º angle and length of 2 sides?

Question

Hi!

So the triangle is very thin, like a spike. It has a 4º angle from its point and has sides of 5.5 feet each. What is the formula for finding that short-length missing side?

Please help :-D

Answer 1

either use the law of cosines or law of sines

for law of cosines

s^2 = 5.5^2 + 5.5^2 -2(5.5)(5.5)cos4

It's a generalized version of the pythagorean theorem when it's not a right angle.

the missing side squared = 5.5^2 + 5.5^2 - 2(5.5)(5.5)cos4

= 2(5.5^2)(1 - cos4) = 2(30.25)(1-.99756) = 60.5(.00244) = 0.1474

the missing side = sqr(0.1474) = 0.384 feet = about 4.61 inches

for law of sines

the triangle is has angles 4 and (180-4)/2 = 88 and 88.

ratios of the side to its opposite angle is a constant

5.5/sin88 = s/sin4 where s= the missing short side

s = 5.5sin4/sin88 = .384 feet = about 4.61 inches

if you get the same answer working the problem two different ways, you have great confidence it's correct

another method is bisect the 4 degree angle and construct a straight line perpendicular to the missing side, s.

you now have two right triangles, each with angles 2-88-90 and with hypotenuse = 5.5. half of s is the base of the right triangle

s/2/5.5 = s/11 = cos88

s = 11cos88 = 0.384 feet= 4.61 inches

work a problem 3 different ways and get the same answer to 3 decimal places, get the same answer, and you're nearly 100% certain of the answer.

Answer 2

Label the third side "d".

d² = (5.5)² + (5.5)² - 2(5.5)(5.5) cos 4°