Question

What is the unit circle value of tan 120, 135, and 150 degrees?

Answer 1

`tan 120^@ = -sqrt(3)`
`tan 135^@ = -1`
`tan 150^@ = -sqrt(3)/3`

Explanation:

Use `tan theta = sin(theta)/cos(theta)`

From a trig circle or a `30^@-60^@-90^@` triangle in the second quadrant:
`tan 120^@ = (sqrt(3)/2)/(-1/2) = sqrt(3)/2 *-2/1 = -sqrt(3)`

From a trig circle or a `45^@-45^@-90^@` triangle in the second quadrant:
`tan 135^@ = (sqrt(2)/2)/(-sqrt(2)/2) = sqrt(2)/2 * -2/sqrt(2) = -1`

From a trig circle or a `30^@-60^@-90^@` triangle in the second quadrant:
`tan 150^@ = (1/2)/(-sqrt(3)/2) = 1/2 * -2/sqrt(3) = -1/sqrt(3) = -1/sqrt(3) * sqrt(3)/sqrt(3) = -sqrt(3)/3`

Answer 2

`color(blue)(rArrtan(120^circ)=-sqrt(3)`

`color(orange)(rArrtan(135^circ)=-1`

`color(purple)(rArrtan(150^circ)=-(sqrt(3))/3`

Explanation:

Let's use the unit circle to find the values

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`color(blue)(tan(120^circ)`

We have the values of `sin(120^circ) and cos(120^circ)`

So, use the identity

`color(brown)(tan(theta)=( sin(theta))/(cos(theta))`

`rarrtan(120^circ)=(sin(120^circ))/(cos(120^circ))`

`rarrtan(120^circ)=(sqrt(3)/2)/(-1/2)`

`rarrtan(120^circ)=sqrt(3)/2*-2/1`

`rarrtan(120^circ)=-cancel2sqrt(3)/cancel2`

`color(blue)(rArrtan(120^circ)=-sqrt(3)`

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`color(orange)(tan(135^circ)`

`color(brown)(tan(theta)=( sin(theta))/(cos(theta))`

`rarrtan(135^circ)=(sin(135^circ))/(cos(135^circ))`

`rarrtan(135^circ)=(sqrt(2)/2)/(-sqrt(2)/2)`

`rarrtan(135^circ)=2/sqrt2*-2/sqrt(2)`

`rarrtan(135^circ)=-cancel((2sqrt2)/(2sqrt2)`

`color(orange)(rArrtan(135^circ)=-1`

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`color(purple)(tan(150^circ)`

`color(brown)(tan(theta)=( sin(theta))/(cos(theta))`

`rarrtan(150^circ)=(sin(150^circ))/(cos(150^circ))`

`rarrtan(150^circ)=(1/2)/(-sqrt(3)/2)`

`rarrtan(150^circ)=1/2*-2/sqrt(3)`

`rarrtan(150^circ)=-cancel2/(cancel2sqrt(3))`

`rarrtan(150^circ)=-1/3`

Here is the most important part. The denominator is an irrational number, so multiplu both numerator and denominator by `sqrt3`

`rarrtan(150^circ)=-1/3*(sqrt(3)/sqrt(3))`

`color(purple)(rArrtan(150^circ)=-(sqrt(3))/3`

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Hope this helps!!! :)