Find the value of 𝑋, given that tangent of 𝑋 over four is equal to the square root of three, where 𝑋 over four is an acute angle.
Since we’re given that this as an acute angle and we know that the tangent of this angle is square root of three, there is a relationship of sides of a right triangle, where square root three is included. So suppose we have this right triangle.
For a 30-60-90 triangle, so the angles of the triangle are 30 degrees, 60 degrees, and 90 degrees, the ratio of their sides 𝐵𝐶 to 𝐴𝐵 to 𝐴𝐶 is one to square root three to two. So 𝐵𝐶 is one, 𝐴𝐵 is square root three, and 𝐴𝐶 is two.
So it states the tangent of 𝑋 over four is square root of three. Well, the tangent of an angle 𝜃 is equal to the opposite side of the angle divided by the adjacent side of the angle. So if we want square root three, we could have square root three over one. So if we want square root three to be our opposite side and one to be our adjacent side, that means our angle would have to be 60 degrees, because square root three is the opposite side and the adjacent side of one is next to 60. Two would be the hypotenuse, which we’re not using.
So 60 is indeed an acute angle, so that’s good. So 60 is our actual angle, so we can set 𝑋 divided by four equal to 60. And now to solve for 𝑋, which we’re- is what we’re trying to find, the value of 𝑋, we can multiply both sides by four. Therefore, 𝑋 is equal to 240 degrees.