# A triangle has sides A, B, and C. The angle between sides A and B is #(7pi)/12#. If side C has a length of #1 # and the angle between sides B and C is #pi/12#, what is the length of side A?

##### 1 Answer

The length of side

#### Explanation:

You can use the law of sines.

The angle between sides

Similarly, the angle between sides

Even though we don't need it for this particular task, let's also call the last remaining angle

According to the law of sines, the following relation between the sides and the opposite angles exists:

#A / sin alpha = B / sin beta = C / sin gamma#

In our case, we only need to look at

#A / sin alpha = C / sin gamma#

#A / sin (pi / 12) = 1 / sin ((7 pi)/12)#

#<=> A = sin (pi / 12) / sin ((7pi) / 12)#

So, the only thing left to do is compute

Let me show you how to do this without the calculator but with some basic knowledge about

#sin (a-b) = sin(a)cos(b) - cos(a)sin(b)# .

You need to express

#sin (pi/12) = sin (pi/3 - pi/4)#

# = sin(pi/3) cos(pi/4) - cos(pi/3)sin(pi/4)#

# = sqrt(3)/2 * 1/sqrt(2) - 1 / 2 * 1 / sqrt(2)#

# = 1/(2 sqrt(2)) (sqrt(3) - 1)#

Similarly,

#sin((7 pi)/12) = sin(pi/3 + pi/4)#

# = sin(pi/3) cos(pi/4) + cos(pi/3) sin(pi/4)#

# = sqrt(3) / 2 * sqrt(2) / 2 + 1 / 2 * sqrt(2) / 2#

# = 1/(2 sqrt(2)) (sqrt(3) + 1)#

Thus, the length of the side

# A = sin (pi / 12) / sin ((7pi) / 12)#

# = (1/(2 sqrt(2)) (sqrt(3) - 1) )/(1/(2 sqrt(2)) (sqrt(3) + 1))#

# = (sqrt(3) - 1) / (sqrt(3) + 1)#

# = ((sqrt(3) - 1) color(blue)((sqrt(3)-1))) / ((sqrt(3) + 1) color(blue)((sqrt(3)-1)))#

# = (sqrt(3) -1)^2 / ((sqrt(3))^2 - 1^2 ) #

# = (3 - 2sqrt(3) + 1) / 2 #

# = 2 - sqrt(3) #