Monday
March 27, 2017

Post a New Question

Posted by on .

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"

Also Give checks.

I know the use of logarithms is unnecessary but i have to show my solution by using logarithms and that's the only reason I'm asking for help. Logarithms confuse me. Please help me.

  • trigonometry - ,

    If you are not already familiar with the law of tangents, here's an article that can help you:

    http://en.wikipedia.org/wiki/Law_of_tangents

    We will be using the same notations in the following solution.

    In the given case, sides a,b are known, and the included angle C. So the sum of angles
    A+B=180-C = 180 - 32-28-30 = 147-31-30 = 1967π/2400 radians
    a+b=21.46+46.28=67.74
    a-b=21.46-46.28=-24.82

    By the law of tangents,
    tan(A-B)/tan(A+B)=(a-b)/(a+b)
    or
    tan(A-B)=tan(A+B)(a-b)/(a+b)
    =0.2331984

    In log (to base 10), cannot calculate the values of negative numbers, so we will keep track of the sign ourselves:
    log(-tan(A+B)) = -0.1962309
    log(-(a-b)) = 1.394801777162711
    log(a+b) = 1.830845192308612
    log(tan(A+B)*(a-b)/(a+b) = -0.1962309+1.3948018-1.8308452
    =-0.6322743
    Antilog(-0.6322743)=0.2331985 as before.
    Convert 0.2331985 to degrees,
    A-B=13-21-41, and
    A+B=147-31-30
    A = (160-53-11)/2=80-26-36
    B = (134-09-49)/2=67-04-55
    side c can be found by the cosine rule or the sine rule.
    Check my work.

  • trigonometry - correction - ,

    Anon & Drws, thank you for pointing out there is a problem with this solution.

    The tangent rule formula that I used was not correct. The formula used for the tangent rule is simply:
    (a-b)/(a+b) = tan((A-B)/2)/tan((A+B)/2)

    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

    Now apply the tangent rule:
    tan((A+B)/2)=3.433633
    tan((A-B)/2)=tan((A+B)/2)*(a-b)/(a+b)
    = -1.2580863
    (A-B)/2 = atan(1.2580863)= -51.520285
    A-B = -103.04057

    A=(147.525-103.04057)/2=22.242215°
    B=(147.525+103.04057)/2=125.282785°

    c can be found by the sine rule or the cosine rule, since 5 of the six unknowns have been calculated. I get c=30.44083 using the cosine rule and it checks with the sine rule.

    I believe you can now handle the logarithm part. If you need further help, just post.

    I apologize again for the unforgivable mistake in applying the tangent rule formula.

Answer This Question

First Name:
School Subject:
Answer:

Related Questions

More Related Questions

Post a New Question