Evaluating inverse trigonometric functions Without using a cal...

Question

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.

$an^{-1}\sqrt3$

Accepted Answer

Hello. Today we're going to be finding the values of tangent inverse of 1 divided by square root of 3 and we're going to write our answer in terms of pi. So the first thing we can do is we can take our inverse trigonometry function, which again is tangent inverse of 1 divided by the square root of 3, and we want to set it equal to a variable which in this case we can allow to be x. Now, since we are working with tangent inverse, we only want the values of x that lie in the region of 0 and pi on the unit circle. The region between 0 and pi of the unit circle is going to be the upper region of the unit circle. So our answers of x should only be the angles that are found in the upper section of the unit circle. So, how can we approach this problem? Well, we can start by simplifying the problem by eliminating the inverse trigonometric function. In order to cancel the tangent inverse on the left hand side, we can take tangent of both sides of the equation. This will allow us to cancel tangent inverse on the left hand side of the equation and we can rewrite the equation as the tangent of x is equal to 1 divided by the square root of 3. So what this is asking us is what angle of x can we plug in a tangent that will give us the value 1 divided by the square root of 3? Well, that angle is going to be pi over 6 in the first quadrant of the unit circle. Now the terminal point at pi divided by 6 is given to us as the square root 3 divided by 2, comma, 1 divided by 2. And recall that tan is equal to the Y value of the terminal point divided by the X value of the terminal point. So if you wanted to double check that this angle is the correct angle for X, we can take the y value, which is 1 divided by 2, and divide that value by square root 3 divided by 2. In order to simplify this, we can take the reciprocal of the denominator and multiply it to both the numerator and denominator of this complex fraction. This will allow us to eliminate the denominator to just 1 leaving us with 1 divided by 2 multiplied by 2 divided by the square root of 3. This will simplify to leave us with the value of 1 divided by the square root of 3, which verifies that the angle pisix when plugged into tangent will give us the output of 1 divided by the square root of 3. So the answer for x is going to be pi divided by 6. And with that being said, the answer to this problem is going to be a. So I hope this video helps you in understanding how to approach this problem, and I'll go ahead and see you all in the next video.

Evaluating inverse trigonometric functions Without using a calculator, evaluate the following expressions.
$\tan^{-1}\sqrt3$

Key Concepts

Inverse Trigonometric Functions

Special Angles in Trigonometry

Quadrants and Angle Values

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