Question

What is the value of tan(sin^-1(1/2))

Answer 1

let's use the angle sum formulas for sin and cos, and the definition that tan = sin / cos

tan (a + b) = sin (a + b) / cos (a + b)
sin (a + b) = [sin a cos b + cos a sin b]
cos (a + b) = [cos a cos b - sin a sin b]

let a = sin^-1(-sqrt(3) / 2), so sin a = -sqrt(3) / 2
at this angle (in Q4), cos a = 1/2 (using 30-60-90 triangle, or the fact that sin^2 + cos^2 = 1)

let b = cos^-1 (-sqrt(3) / 2), so cos b = -sqrt(3) / 2
at this angle (in Q2), sin b = 1/2

thus, tan (a + b) = [-sqrt(3)/ 2 (-sqrt(3)/ 2) + (1/2)(1/2)] / [-sqrt(3) / 2 (1/2) - (-sqrt(3) / 2)(1/2)]

tan (a + b) = [3/4 + 1/4] / [-sqrt(3) / 4 + sqrt(3) / 4]
tan (a + b) = 1 / (0), and is thus undefined

tan (sin^-1(-sqrt(3) / 2) + cos^-1(-sqrt(3) / 2)) is undefined

this is the same answer I got using the angle sum formula for tan(a + b) = (tan a + tan b) / [1 - tan a tan b]
sin a = -sqrt(3) / 2 ==> tan a = -sqrt(3)
cos b = -sqrt(3) / 2==> tan b = -sqrt(3) / 3
(tan a + tan b) / (1 - tan a tan b)
denominator = 1 - tan a tan b = 1 - (-sqrt(3))(-sqrt(3) / 3)
= 1 - 3/3 = 1 - 1 = 0
tan (a + b) is undefined

let's verify with the angles (since we can)
sin^-1 (-sqrt(3) / 2) = -60
cos^-1 (-sqrt(3) / 2) = 150
150 - 60 = 90, and yes, tan 90 is undefined

Answer 2

https://answers.yahoo.com/question/index;_ylt=A0LEV7lYwolU41EAoKgPxQt.;_ylu=X3oDMTByMG04Z2o2BHNlYwNzcgRwb3MDMQRjb2xvA2JmMQR2dGlkAw--?qid=20111214094447AAWygca

Answer 3

let \(\theta=\sin^{-1}(1/2)\) then \(\sin(\theta)=1/2\)

draw a right triangle

Answer 4

use Pythagorean theorem to get the 3rd side

Answer 5

\(\tan(\theta)\) is the opposite side over the adjacent side

therefore \[\tan(\sin^{-1}(1/2))=\tan(\theta)=\frac{1}{\sqrt{3}}\]