How do you find the exact value of tan(sin−1(23))?

1 Answer
Bill K.
Jul 16, 2015

Since tan(x)=sin(x)cos(x), we must find sin(sin−1(23)) and cos(sin−1(23)). Clearly, sin(sin−1(23))=23 by the definition of an inverse function (in this case, to be more precise, sin(sin−1(x))=x for all −1≤x≤1). Also, the cosine of the "angle" sin−1(23) is positive, therefore by the Pythagorean identity, cos(sin−1(23))=√1−sin2(sin−1(23))
=√1−(23)2=√59=√53.

(You could also draw a right triangle, label one of the angles sin−1(23), label the side lengths appropriately, use the Pythagorean theorem and SOH, CAH, TOA appropriately to do this last calculation.)

Therefore, tan(sin−1(23))=23√53=2√5=2√55

Related questions
See all questions in Basic Inverse Trigonometric Functions
Impact of this question
44283 views around the world
You can reuse this answer
Creative Commons License