If x is an acute angle, and tan x /3/4, evalua
To evaluate the given expression, we can start by recalling the definition of the tangent function. The tangent of an angle is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle in a right triangle.
Given that tan(x) = 3/4, we can construct a right triangle where the side opposite x has a length of 3, and the side adjacent to x has a length of 4.
Once the triangle is constructed, we can use the Pythagorean theorem to find the length of the hypotenuse. The Pythagorean theorem states that the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides in a right triangle.
Using the lengths of the two sides, we can calculate the length of the hypotenuse as follows:
hypotenuse^2 = opposite^2 + adjacent^2
hypotenuse^2 = 3^2 + 4^2
hypotenuse^2 = 9 + 16
hypotenuse^2 = 25
Taking the square root of both sides, we find:
hypotenuse = sqrt(25)
hypotenuse = 5
Therefore, in the constructed right triangle, the hypotenuse has a length of 5.
Finally, we can evaluate the expression tan(x) using the tangent ratio:
tan(x) = opposite / adjacent
tan(x) = 3 / 4
So, the value of x, where x is an acute angle, such that tan(x) = 3/4, is determined by arctan(3/4) or using a calculator, which gives approximately 36.9 degrees.