SOLUTION: Please help me solve this question: If A is an acute angle such that SinA = {{{2 sqrt(2)/3}}} calculate without a calculator (1) cosA (2) sin2A (3) Tan2A I tried to first get

Question 961988: Please help me solve this question: If A is an acute angle such that SinA = calculate without a calculator (1) cosA (2) sin2A (3) Tan2A
I tried to first get the reference angle and solve it by finding a general solution but then I remembered that I can't use my calculator.

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
well, you can't use a calculator to solve the problem, but you can use a calculator to see if you solved it correctly.
they didn't say you couldn't do that.

first we'll solve without the use of the calculator.

you are given that sinA = 2*sqrt(2)/3

draw yourself a triangle and call it ABC with your angle in question at A.
since sine is opposite / hypotenuse, then opp/hyp = 2*sqrt(2)/3.

this says that opp = 2*sqrt(3) and h = 3 because opp/hyp = 2*sqrt(3) / 3.

use the pythagorean formula to solve for adj.

you will find that adj is equal to 1.

your triangle has:

hyp = 3
opp = 2 * sqrt(2)
adj = 1

you get sinA = opp/hyp = 2 * sqrt(2) / 3

you get cosA = adj/hyp = 1/3

you get tanA = opp/adj = 2*sqrt(2).

you now want to find sin2A.

the formula for that is sin2A = 2*sinA*cosA

that becomes sin2A = 2*2*sqrt(2)/3*1/3 which becomes sin2A = 4*sqrt(2)/9.

you now want to find tan2A.

the formula for that is tan2A = (2*tanA) / (1-tan^2(A)).

that becomes tan2A = 2*2*sqrt(2) / (1-(2*sqrt(2))^2) which becomes:

tan2A = 4*sqrt(2) / (1 - 4*2) which becomes:

tan2A = 4*sqrt(2) / -7.

your solutions are:

sinA = 2*sqrt(2)/3
cosA = 1/3
tanA = 2*sqrt(2)
sin2A = 4*sqrt(2)/9
tan2A = 4*sqrt(2)/(-7)

you can confirm using your calculator.
first find the angle and then calculate the functions and then compare those functions with what you have here.

they should match.

the calculator will give you decimal equivalents.
just find the decimal equivalents of what you have here and you'll get your match.

for example:

sinA = 2*sqrt(2)/3.
solve for A to get A = 70.52877937 degrees.

tan2A = 4*sqrt(2)/(-7)

use the calculator to get tan(2*70.52877937) = tan(141.0575587) = -.8081220356

use the calculator to calculate 4*sqrt(2)/(-7) and you will find that the calculator tells you that the value is -.8081220356.

they match so the solution is good.

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