Question

A triangle has sides A, B, and C. The angle between sides A and B is `(3pi)/4`. If side C has a length of `16` and the angle between sides B and C is `pi/12`, what are the lengths of sides A and B?

Answer 1

Side `b = 11.31` and Side `a = 5.86`
:)

Explanation:

We can get the value of side A and B by using the "Law of Sines",

`sin A/a = Sin B/b = Sin C/c`

First, we must convert the radian value to degree value.

To convert radian value to degree value,

Multiply it by `180/pi`

since Angle `A = pi/12`

`A =pi/12 * 180/pi` = `(180pi)/(12pi)` = `(180cancelpi)/(12cancelpi)`

Angle `A = 15^o`

and Angle `C = (3pi)/4 or 135^o`

`C = (3pi)/4 * 180/pi` = `(540pi)/(4pi)` = `(540cancelpi)/(4cancelpi)`

Angle `C = 135^o`

using the law of sines,

`sin A/a = sin C/c`

`(sin15^o)/a = (sin135^o)/16`

using algebraic technique we get,

`a = ((sin15^o)(16))/(sin135^o)`

`a = 5.8564`

we use again the "Law of Sines", since "Pythagorean Theorem" doesn't work on Non-Right Triangles,

since angle `B` is unknown, we can get its value by simply getting the difference of `180^o-(`Angle`C +`Angle`A)`, since "the sum of all interior angles of a triangle is always `180^o`"

Angle `B = 180^o-` `(135^o` `+15^o)`

`= 180^o-150^o`

Angle `B = 30^o`

Applying again the Law of Sines to get the value of side of `b`, we get,

`sin B/b = sin C/c`

`(sin30^o)/b = (sin135^o)/16`

Applying algebraic technique, we get,

Side `b = ((sin30^o)(16))/(sin135^o)`

Hence, we get:

Side `b = 11.3137`

Tip on Trigonometry:
Pythagorean Theorem is only reliable in solving right triangles, while Law of Sines and Cosines works in almost any triangles

:)