plz help trigonometry?

Question

plz show work so i can understand. thanks


Note that the point value of each question is given in parentheses and give checks for the following question.

9. (10) Use logarithms and the law of tangents to solve the triangle ABC, given that a= 21.46 ft, b= 46.28 ft, and c= 32º 28’ 30”

Answer 1

Wow!!! I'd never heard of the Law of Tangents, and a recently published trig textbook doesn't mention such a thing. But I have a 1954 high school text that says this:

"The Law of Tangents: In any triangle the difference of any two sides is to their sum as the tangent of half the difference of the angles opposite these sides is to the tangent of half their sum."

It goes on to say, "Given two sides and the included angle, the law of tangents is preferable to the law of cosines ... when large numbers are involved and when it is desired to use logarithms."

Algebraically, the Law of Tangents says,

(a-b)/(a+b) = tan[(A-B)/2] / tan[(A+B)/2]

In your problem, b>a, so we'll use the "b-a" form of the Law, rather than "a-b". We have

a = 21.46 b = 46.28 C = 32d 28m 30s

b+a = 67.74 b-a = 24.82
(B+A)/2 = (180-C)/2 = 90d - (16d 14m 15s) = 73d 45m 45s (Eq 1)

From the Law of Tangents:

tan[(B-A)/2] = (b-a) tan[(B+A)/2] / (b+a)
tan[(B-A)/2] = 24.82 tan(73d 45m 45s) / 67.74
log tan[(B-A)/2] = log 24.82 + log tan(73d 45m 45s) - log 67.74
log tan[(B-A)/2] = 1.39480 + 0.53575 - 1.83085 = 0.09971
(B-A)/2 = 51.520285 = 51d 31m 13s (Eq 2)

Solving Equations 1 and 2, we have

B = 125d 16m 58s A = 22d 14m 32s (Answers)

(noting that A+B+C = 180)

Finally get c from the Law of Sines:

c / sin C = a / sin A
c = a sin C / sin A = 21.46 x sin(32d 28m 30s) / sin(22d 14m 32s)
log c = log 21.46 + log sin(32d 28m 30s) - log sin(22d 14m 32s)
log c = 1.33163 + (-0.27008) - (-0.42191) = 1.48346
c = 30.44 (Answer)

Check using Law of Sines:

sin A / a = sin B / b = sin C / c
0.0176385 = 0.0176385 = 0.0176390

That's close enough using "calculator accuracy."

Answer 2

@Josephine thanks for the free textbook! On-topic: the hardest trigonometry questions are once you're the guy who ought to provide your self a question. operating example, attempt to coach some thing that isn't any longer widely used to mankind, or a minimum of no longer widely used to you.

Answer 3

Do you have a TI83+ calc?

You need one to use the "tan" function.