How to find the value of
arctan(2 sin (pi/3))
without using calculator or sketching a triangle. It is like taking u=something and then evaluating.
Thanks.
the solution relies on the fact that you should know the ratio of sides of the 30,60 90 degree triangle, and knowing that Pi/3 = 60 and pi/6 = 30º
in pi/3 = √3 /2 so 2 sin pi/3 = √3 or √3 /1
now what is the angle so that tan(angle) = √3 ???
must be 60º or in radians pi/3
To find the value of arctan(2 sin (pi/3)) without using a calculator or sketching a triangle, we can make use of the properties of trigonometric functions and the knowledge of special angles.
1. Start by recognizing that pi/3 is equivalent to 60 degrees. This means that we need to find the angle that has a tangent equal to the square root of 3.
2. Recall the definition of the tangent function: tan(x) = opposite/adjacent. In this case, we want to find the angle whose tangent is equal to the square root of 3.
3. One way to approach this is by considering the special right triangle with angles 30 degrees, 60 degrees, and 90 degrees. In this triangle, opposite/adjacent = tan(30). Since the ratio opposite/adjacent is the same as sine, we can say that sin(30) = 1/2. Therefore, tan(30) = sin(30)/cos(30) = (1/2) / (√3/2) = 1/√3 = √3/3.
4. Now, we need to find the angle whose tangent is equal to the square root of 3. Since we know that tan(30) = √3/3, we can conclude that the angle we are looking for is 30 degrees (pi/6 radians).
Therefore, the value of arctan(2 sin (pi/3)) is pi/6 radians or approximately 30 degrees, without the need for a calculator or drawing a triangle.