Question

How do you find the exact value of `tan(cos^-1 (1/2))`?

Answer 1

tan (pi/3) = sqrt3
tan (-pi/3) = - sqrt3

Explanation:

`cos ^-1 (1/2)` --> `arccos (1/2)`
Trig table and unit circle -->
`cos x = 1/2` --> arc `x = +- pi/3`
There are 2 answers for `(0, 2pi)`
`tan (pi/3) = sqrt3`
`tan (-pi/3) = - sqrt3`

Answer 2

see below

Explanation:

From the `color(red)(cos^-1(1/2)` we have the side `color(red)(adjacent` to `color(red)(color(red)(angle theta=1` and the side `color(red)(hypote n use = 2` . So this is a `color(red)(30^@-60^@-90^@` triangle and `color(red)(theta = 60^@`therefore the `color(red)(opposite` to `color(red)(angle theta = sqrt3`.

Note that from the restrictions for the range of inverse circular functions `color(red)(cos^-1 x` is restricted to quadrants 1 and 2 and since the argument is positive `color(red)(1/2)` our answer will be in quadrant 1 only.

Hence,
`color(blue)(tan(cos^-1(1/2))=tan theta=(opposite)/(adjacent)=sqrt3/1=sqrt3`