Question

A triangle has sides A, B, and C. Sides A and B are of lengths `12` and `5`, respectively, and the angle between A and B is `pi/12`. What is the length of side C?

Answer 1

`169-30*sqrt2*(sqrt3+1)`

Explanation:

We use Cosine-rule for the given triangle & get

`C^2`=`A^2+B^2-2AB(cosangle betwn. A & B)`
`=12^2+5^2-2*12*5*cos(pi/12)`
`=144+25-120{sqrt(1/2(1+cos(2*pi/12))}`
`=169-120{sqrt(1/2(1+sqrt3/2))}`
`=169-120{sqrt{(2+sqrt3)/4)}`
`=169-120/2{sqrt((2+sqrt3))}`
`=169-60(sqrt(2+sqrt3))`
`=169-60(sqrt((4+2sqrt3)/2)`
`=169-(60/sqrt2)*{sqrt(3+1+2sqrt3)}`
`=169-(60*sqrt2)/2*[sqrt{(sqrt3)^2+(sqrt1)^2+2*sqrt3*sqrt1}]`
`=169-30*sqrt2*sqrt[(sqrt3+sqrt1)^2]`
`=169-30*sqrt2*(sqrt3+1)`

Answer 2

`color(red)("A different method; just to demonstrate that you can")`

I have taken you to the point where all you have to do is 'plug' the values into a calculator.

Explanation:

Always a good start to draw a diagram or sketch.

`color(blue)("Method Plan")`

Objective: determine `/_b` then by using the `sin(/_b)` determine C
'............................................

Step 1: `->` determine h; then cP using `sin(/_c)`

Step 2:`->` 12 - cP = bP

Step 3:`-> /_b = arctan(h/(bP))`

Step 4: `=> sin(/_b) = h/C " "=>" " C=h/sin(/_b)`

To reduce accumulated rounding error it is better do the calculation only at the end.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question")`

Step 1: `" "color(brown)(h=Bsin(/_c)->5sin(pi/12))`

`" "color(brown)(cP = Bcos(/_c) -> 5cos(pi/12))`
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Step 2:`" "color(brown)(bP=12-5cos(pi/12))`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Step 3: `" "color(brown)( /_b = arctan(h/(bP))->/_b=arctan((5sin(pi/12))/(12-5cos(pi/12))))`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Step 4: `" "color(brown)(C=h/sin(/_b)->color(blue)(C=h/[sin[arctan((5 sin(pi/12))/(12-5cos(pi/12))) ]))`

`color(green)("You will see arctan on your calculator as "tan^(-1))`
Arctan returns angle that made that made the tangent value

So if `tan(theta)=x" "` then `" "arctan(x)=theta`