SOLUTION: Solve triangle ABC. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠A1 is smaller than ∠A2.) b = 128,

Algebra ->  Trigonometry-basics -> SOLUTION: Solve triangle ABC. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that ∠A1 is smaller than ∠A2.) b = 128,       Log On


   

Question 1159444: Solve triangle ABC. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that
∠A1 is smaller than ∠A2.)
b = 128, c = 162, ∠B = 46°
Found 2 solutions by MathLover1, MathTherapy:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Solve triangle ABC. (If an answer does not exist, enter DNE)
< A%5B1%5D is smaller than < A%5B2%5D
b+=+128, c+=+162, < B+=+46°

use the Cosine Rule

cos%28B%29+=++%28c%5E2+%2B+a%5E2+-+b%5E2%29%2F%282ca%29
cos%2846%29+=++%28162%5E2+%2B+a%5E2+-+128%5E2%29%2F%282%2A162a%29
0.694658370459=++%2826244+%2B+a%5E2+-+16384%29%2F%28324a%29
324a%2A0.694658370459=+a%5E2+%2B+9860
225.07a=+a%5E2+%2B+9860
+a%5E2-+225.07a%2B+9860=0-> use quadratic formula
a=+%28-%28-225.07%29+%2B-+sqrt%28+%28-225.07%29%5E2-4%2A1%2A9860+%29%29%2F2+
a+=+%28225.07+%2B-+sqrt%28+11216.5049%29%29%2F2+
a+=+%28225.07+%2B-+105.91%29%2F2+
solutions:
a+=+%28225.07+%2B+105.91%29%2F2+
a=+165.5
or
a+=+%28225.07+-+105.91%29%2F2+
a+=59.6


< A%5B1%5D if a+=59.6

cos%28A%29+=++%28b%5E2+%2B+c%5E2+-+a%5E2%29%2F%282bc%29
cos%28A%29+=++%28128%5E2+%2B+162%5E2+-+59.6%5E2%29%2F%282%2A128%2A162%29
cos%28A%29+=0.9422222222222221
A=cos%5E-1%280.9422222222222221%29
A%5B1%5D=19.57°

< A%5B2%5D if a+=165.5
cos%28A%29+=++%28128%5E2+%2B+162%5E2+-+165.5%5E2%29%2F%282%2A128%2A162%29
cos%28A%29+=+0.3674225983796296
A+=+cos%5E-1%280.3674225983796296%29
< A%5B2%5D=68.44°

use both values of angle A to find which angle C will be your solution:
< C=180-%28A%5B1%5D%2BB%29
<+C=180-%2819.572%2B46%29
< C=114.43°
or
< C=180-%28A%5B1%5D%2BB%29
< C=180-%2868.44%2B46%29
< C=65.56+°

so, your solution is:
a=59.6
b+=+128+
c+=+162
< A=19.57°
< B+=+46°
< C=114.43°
or
a=165.5
b+=+128+
c+=+162
< A=68.44°
< B+=+46°
< C=65.56+°


Answer by MathTherapy(10555) About Me  (Show Source):
You can put this solution on YOUR website!
Solve triangle ABC. (If an answer does not exist, enter DNE. Round your answers to one decimal place. Below, enter your answers so that
∠A1 is smaller than ∠A2.)
b = 128, c = 162, ∠B = 46°
Unless of course, you feel like TORTURING yourself, then you can go by the other person's more complex, BORING, and time-consuming method.

Otherwise, this is the correct way to do this! 

To find ∡C, use LAW of SINES, or: 

After solving, you'll  find ∡C to be 65.5626863o, or about 66o

Now, as sin is > 0 in the 2nd quadrant, ∡C can ALSO be 180o - 66o, or 114o.



If ∡C is 66o, and with ∡B being 46o, then ∡A MUST be 68o (180o - 66o - 46o).

If ∡C is 114o, and with ∡B being 46o, then ∡A MUST be 20o (180o - 114o - 46o).

Now, it's given that ∡A1 < ∡A2, then it goes without saying that ∡A MUST measure the SMALLER of the 2, or 20o. 

We now know that: 

You can now use any of the above angles as well as any corresponding side, along with the LAW of SINES, to get side a.

In other words, 

That's IT!!