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.
