SOLUTION: In triangle ABC, AB=24m and angle ABC=90degree. Given that sin angle ACB=3/5, find I) the length of AC II) the length of BC III) the value of cos angle ACB+ tan angle BAC

Algebra ->  Trigonometry-basics -> SOLUTION: In triangle ABC, AB=24m and angle ABC=90degree. Given that sin angle ACB=3/5, find I) the length of AC II) the length of BC III) the value of cos angle ACB+ tan angle BAC      Log On


   

Question 1120749: In triangle ABC, AB=24m and angle ABC=90degree. Given that sin angle ACB=3/5, find
I) the length of AC
II) the length of BC
III) the value of cos angle ACB+ tan angle BAC
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
triangle ABC is a right triangle with the right angle being at angle ABC.

AB = 24 meters.

sin(ACB) = 3/5.

since sine is opposite / hypotenuse, then sin(ACB) = 3/5 = 24/AC

since 3/5 = 24/AC, solve for AC to get AC = 24 * 5 / 3 = 40.

by pythagorus, [AB]^2 + [BC]^2 = [AC]^2 which becomes 24^2 + [BC]^2 = 40^2.

solve for BC to get BC = sqrt(40^2 - 24^2) = 32.

you have AB = 24, AC = 40, BC = 32.

cos(ACB) = adjacent / hypotenuse = BC/AC = 32/40

tan(BAC) = opposite / adjacent = BC / AB = 32/24

cos(ACB)) + tan(BAC) = 32/40 + 32/24 = 4/5 + 4/3 = 12/15 + 20/15 = 32/15.

that should be your answer.

solutions are:

AB = 24
AC = 40
BC = 32
sin(ACB) + tan(BAC) = 32/15.

you could solve by finding the angles involved but that was unnecessary.

for example:

sin(ACB) = 3/5 leads to angle ACB = 36.86989765.

since sum of angles of triangle is 180, then BAC = 180 - 90 - 36.86989765 = 53.13010235.

you have:
angle ABC = 90
angle ACB = 36.86989765
angle BAC = 53.13010235.

sin(ACB)) = c/b becomes sin(36.8689765) = 24/b

solve for b to get b = 24/sin(36.8689765) = 40

cos(ACB) = a/b bcomes cos(36.8689765) = a/40

solve for a to get a = 40 * cos(36.8689765) = 32.

cos(ACB) + tan(BAC) = cos(36.86989765 + tan(53.13010235) = 2.133333333....

the denominator is assumed to be 1.

multiply numerator and denominator of that by 15 and it becomes 32/15.

that'as the same answer i got earlier.

all answser check out.

your solutions are:

AC = 40
BC = 32
cos(ACB) + tan(BAC) = 32/15 = 2.3333333333......

my diagram is shown below.

in the diagram:

c is the same as AB
b is the same as AC
a is the same as BC.

angles shown are rounded to 1 decimal point, but calculations are made from the maximum number of decimal points for each as stored in the calculator.

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