Question

How do you find the value of `tan(pi/3)`?

Answer 1

`sqrt3`

Explanation:

If you know the values of `sin(pi/3)` and `cos(pi/3)`, you can write that

`tan(pi/3)=sin(pi/3)/cos(pi/3)=(sqrt3/2)/(1/2)=sqrt3/2(2/1)=sqrt3`

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Alternatively, you could think of this as `tan(60˚)`, and then draw a `30˚-60˚-90˚` triangle:

`tan(60˚)` will be equal to `"opposite"/"adjacent"` in reference to the `60˚` angle, so we see that `"opposite"=sqrt3` and `"adjacent"=1`. Hence,

`tan(60˚)="opposite"/"adjacent"=sqrt3/1=sqrt3`

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We can also examine the unit circle at `pi/3`:

If we know the point `(1/2,sqrt3/2)`, we can determine tangent if we think about tangent as the slope of the line in the unit circle. Since the line originates at `(0,0)`, its slope is

`tan(pi/3)=(sqrt3/2-0)/(1/2-0)=sqrt3`

This idea of `"slope"=(Deltay)/(Deltax)` is analogous to tangent because the sine values correlate to the `y` values of the ordered pair, and cosine with `x`, so remembering that `tan(x)=sin(x)/cos(x)` and that tangent is slope should be fairly intuitive.

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