Question

How do you find the exact value of `tan300`?

Answer 1

`tan(300^circ)=color(green)(-sqrt(3))`

Explanation:

`300^circ` is equivalent to `(-60^circ)`
A `60^circ` right triangle is one of the standard triangles.

`tan = "opposite"/"adjacent"`

Answer 2

`tan(300^@) = -sqrt(3)`

Explanation:

`300^@=360^@-60^@` equivalently we have
`2pi-pi/3 = 300^@/180^@ pi` then
`tan(300^@)=tan(2pi-pi/3)` but `tan` is periodic with period `pi` so
`tan(2pi-pi/3)=tan(-pi/3)` and `tan` is an odd function so
`tan(-pi/3)=-tan(pi/3)`
Now `tan(x)=sin(x)/cos(x)` and `sin(pi/3)=1/2` so
`cos(x)=sqrt(1-cos^2(x)) = sqrt(1-1/4)=sqrt(3)/2`

Finally

`tan(300^@) = -sqrt(3)`