Evaluate tan(cos^(-1)(�ã3/2 )+tan^(-1)(�ã3/3))?
arccos(sqrt(3) / 2) = pi / 6
arctan(sqrt(3) / 3) = pi / 6
tan(pi / 6 + pi / 6)
= tan(pi / 3)
= sqrt(3).
Evaluate tan(cos^(-1)(sqrt 3)/2))+tan^(-1)((sqrt 3)/3))?
tan(cos^(-1)(sqrt 3)/2))+tan^(-1)((sqrt 3)/3))
= tan( pi/6) + pi/6
= 1/√3 + pi/6
To evaluate the expression tan(cos^(-1)(�ã3/2 )+tan^(-1)(�ã3/3)), we need to first find the values of cos^(-1)(�ã3/2 ) and tan^(-1)(�ã3/3), and then substitute those values into the expression.
1. To find cos^(-1)(�ã3/2 ):
- The inverse cosine function (cos^-1) gives us the angle whose cosine is equal to a given value.
- In this case, we need to find the angle whose cosine is equal to (�ã3/2 ).
- Since the cosine function represents the ratio of the adjacent side to the hypotenuse in a right triangle, let's consider a right triangle with a hypotenuse of 1 and an adjacent side of (�ã3/2 ).
- By applying the Pythagorean theorem, we find that the opposite side is 1/2.
- Therefore, the ratio of the adjacent side to the hypotenuse in this triangle is (�ã3/2 ), which is equal to the cosine of the angle.
- So, cos^(-1)(�ã3/2 ) = pi / 6.
2. To find tan^(-1)(�ã3/3):
- The inverse tangent function (tan^-1) gives us the angle whose tangent is equal to a given value.
- In this case, we need to find the angle whose tangent is equal to (�ã3/3).
- Consider a right triangle with an opposite side of (�ã3) and an adjacent side of 3.
- In this triangle, the ratio of the opposite side to the adjacent side is (�ã3/3), which is equal to the tangent of the angle.
- So, tan^(-1)(�ã3/3) = pi / 6.
Now, we substitute the values we found into the expression:
tan(cos^(-1)(�ã3/2 )+tan^(-1)(�ã3/3))
= tan(pi / 6 + pi / 6)
= tan(pi / 3)
= sqrt(3).
Therefore, the evaluation of the expression is sqrt(3).