law of cosines?

Law of cosines equation is:

c² = a² + b² - 2ab cos(C)

Where the length "c" is opposite of angle "C".

You are given angle "A" and do not have length "a", so we can rearrange the variables to be:

a² = b² + c² - 2bc cos(A)

Now we can substitute what we know and solve for "a":

a² = 17² + 16² - 2(17)(16) cos(84)

a² = 289 + 256 - 544 cos(84)

a² = 545 - 544 cos(84)

a² = 545 - 544(0.1045285)

I rounded here, but didn't in my calculator to reduce errors due to rounding:

a² = 545 - 56.86348

a² = 488.136516

a = 22.093812

I'll keep things to 5 or 6 DP to try to limit rounding errors until the end.

Now that we know the length of "a", we can use law of sines with angle "A" and length "b" to solve for angle "B":

22.093812/sin(84) = 17/sin(B)

22.093812 sin(B) = 17 sin(84)

sin(B) = 17 sin(84) / 22.093812

Again, decimal approximation, but not rounded in my calculator:

sin(B) = 17(0.994522) / 22.093812

sin(B) = 16.906872 / 22.093812

sin(B) = 0.765231

Inverse sine function to get:

B = 49.928° (rounded to 3DP)

law of cosines relates the lengths of the sides of a plane triangle to the cosine of one of its angles. If a, b, c are the three sides of a triangle, and C is the angle between a and b and opposite side c, then:

c² = a² + b² – 2abcosC

or cos C = (a² + b² – c²) / (2ab)

cos A = (b² + c² – a²) / (2bc)

cos B = (a² + c² – b²) / (2ac)

A = 84º

c = 16

b = 17

rearrange the formula so you can get a

cos A = (b² + c² – a²) / (2bc)

b² + c² – a² = (2bc) cos A

a² = b² + c² – (2bc) cos A

a² = 17² + 16² – (2•17•16) cos 84

a² = 289 + 256 – 56.86

a = 22.09

cos B = (a² + c² – b²) / (2ac)

cos B = (22.09² + 16² – 17²) / (2•22.09•16)

cos B = (487.97 + 256 – 289) / (706.88) = 0.6436

B = 49.9º

The cosine of an angle is the ratio between one side and the other side.

So if you have a triangle where the base is 17 and the hypotenuse is 16, the cosine of the angle is 16/17, or 0.9412 (rounded to 4 places).

So your job is to find the angle whose cosine is 0.9412. My trig calculator is at work but you could figure that out. I get 19.7459 degrees.