Use logarithms and the law of tangents to solve the triangle ABC, given that a=21.46 ft, b=46.28 ft, and C=32°28'30"
Using the previous values,
a=21.46
b=46.28
C=32-28-30=32.475
We have
A+B=180-32.475=147.525
a+b=67.74
a-b=-24.82
tan((A+B)/2)=3.433633
tan((A-B)/2)=tan((A+B)/2)*(a-b)/(a+b)
= -1.2580863 <<.. i did this part in logarithms it was actually very easy now that i understood it. thank you!
but i cant seem to understand what u did in the following part .
(A-B)/2 = atan(1.2580863)= -51.520285
A-B = -103.04057
I didn't get what u did there and when i tried it i came up with different numbers. Can you please clarify the steps.
thank you for taking the time to help me. I really appreciate it.
A=(147.525-103.04057)/2=22.242215°
B=(147.525+103.04057)/2=125.282785°
Sorry, it must have been a typo.
It should read:
(A-B)/2 = atan(-1.2580863)= -51.520285
So multiply by 2 to get
A-B = -103.04057
The negative sign was because I did not choose A and B wisely. It turned out that A<B.
So if
A-B=-103.04057....(1)
A+B=147.525.....(2)
Add (1) and (2)
2A+0B=44.484430 => A=22.242215
Subtract (1) from (2)
0A+2B=250.56557 => B=125.282785
Notice that B>A, so A-B is negative, so is arcTan((A-B)/2).
Certainly! I apologize if the steps were unclear. Let me explain the process in more detail.
First, we have found that tan((A+B)/2) = 3.433633. To solve for (A-B)/2, we can use the law of tangents, which states that tan((A-B)/2) = (tan(A/2) - tan(B/2)) / (1 + tan(A/2) * tan(B/2)).
In this case, we have tan((A+B)/2) = 3.433633 and a-b = -24.82. So, we can substitute these values into the law of tangents to get:
tan((A-B)/2) = tan((A+B)/2) * (a-b)/(a+b)
Plugging in the values:
tan((A-B)/2) = 3.433633 * (-24.82)/(21.46+46.28)
= -1.2580863
To find (A-B)/2, we need to use the arctan function (often written as atan or tan^(-1)). So, we take the inverse tangent of -1.2580863:
(A-B)/2 = atan(-1.2580863)
= -51.520285 degrees
Next, we multiply (A-B)/2 by 2 to find A-B:
A-B = -51.520285 * 2
= -103.04057 degrees
Finally, we use the values of (A+B)/2 and (A-B)/2 to find A and B:
A = ((A+B)/2) - ((A-B)/2)
= (147.525 - (-103.04057)) / 2
= 22.242215 degrees
B = ((A+B)/2) + ((A-B)/2)
= (147.525 + (-103.04057)) / 2
= 125.282785 degrees
I hope this explanation clarifies the steps for you. If you have further questions, feel free to ask!