Question 1195245: Given that x is an acute angle and cos x=(2√5)/7 , find without using mathematical tables or a calculator , the value of tan(90-x)°
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! see my worksheet below:
since cos(x) = 2 * sqrt(50 / 7, and cos(x) = adjacent / hypotenuse, then:
adjacent = 2 * sqrt(5) and hypotenuse = 7.
since a^2 + b^2 = c^2, and since a^2 = adjacent squared and since b^2 = opposite squared and since c^2 = hypotenuse squared, then:
a^2 + b^2 = c^2 becomes (2 * sqrt(5)^2 + opposite squared = 7^2
solve for opposite squared to get opposite squared = b^2 = 49 - 20 = 29.
you have side adjacent to x = 2 * sqrt(5), side opposite x = sqrt(29) and hypotenuse = 7.
the angle of 90 - x is the other acute angle of the right triangle formed.
the tangent of that angle is equal to opposite / adjacent which is equal to 2 * sqrt(5) / sqrt(29).
that has been simplified to 2 * sqrt(5) * sqrt(29) / 29 which has been further simplified to 2 * sqrt(145) / 29.
your answer is either 2 * sqrt(5) / sqrt(29) or the more simplified version of 2 * sqrt(145) / 29.
i used my calculator to confirm this was correct.
2 * sqrt(5) / sqrt(29) = .8304547985.
2 * sqrt(145) / 29 = the same.
90 - x = arctan(.8304547985) = 39.7080987 degrees.
arccos(2 * sqrt(5) / 7) = 50.2919013 degrees.
this confirms the angles are complementary, as they should be.
the problem was solved without the use of a calculator or table.
it was, however, confirmed to be correct through the use of a calculator.
Answer by ikleyn(52847)  (Show Source):
You can put this solution on YOUR website! .
Notice that without using tables or a calculator, we can find an exact EXPRESSION for tan(90°-x), ONLY,
but not its value itself.
I am telling it to explain you that the formulation of the problem is incorrect and should be changed.
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