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

Mar 13, 2017

$\tan {120}^{\circ} = - \sqrt{3}$
$\tan {135}^{\circ} = - 1$
$\tan {150}^{\circ} = - \frac{\sqrt{3}}{3}$

#### Explanation:

Use $\tan \theta = \sin \frac{\theta}{\cos} \left(\theta\right)$

From a trig circle or a ${30}^{\circ} - {60}^{\circ} - {90}^{\circ}$ triangle in the second quadrant:
$\tan {120}^{\circ} = \frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}} = \frac{\sqrt{3}}{2} \cdot - \frac{2}{1} = - \sqrt{3}$

From a trig circle or a ${45}^{\circ} - {45}^{\circ} - {90}^{\circ}$ triangle in the second quadrant:
$\tan {135}^{\circ} = \frac{\frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}} = \frac{\sqrt{2}}{2} \cdot - \frac{2}{\sqrt{2}} = - 1$

From a trig circle or a ${30}^{\circ} - {60}^{\circ} - {90}^{\circ}$ triangle in the second quadrant:
$\tan {150}^{\circ} = \frac{\frac{1}{2}}{- \frac{\sqrt{3}}{2}} = \frac{1}{2} \cdot - \frac{2}{\sqrt{3}} = - \frac{1}{\sqrt{3}} = - \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = - \frac{\sqrt{3}}{3}$

Mar 14, 2017

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 $\sin \left({120}^{\circ}\right) \mathmr{and} \cos \left({120}^{\circ}\right)$

So, use the identity

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

$\rightarrow \tan \left({120}^{\circ}\right) = \frac{\sin \left({120}^{\circ}\right)}{\cos \left({120}^{\circ}\right)}$

$\rightarrow \tan \left({120}^{\circ}\right) = \frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}}$

$\rightarrow \tan \left({120}^{\circ}\right) = \frac{\sqrt{3}}{2} \cdot - \frac{2}{1}$

$\rightarrow \tan \left({120}^{\circ}\right) = - \cancel{2} \frac{\sqrt{3}}{\cancel{2}}$

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

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color(orange)(tan(135^circ)

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

$\rightarrow \tan \left({135}^{\circ}\right) = \frac{\sin \left({135}^{\circ}\right)}{\cos \left({135}^{\circ}\right)}$

$\rightarrow \tan \left({135}^{\circ}\right) = \frac{\frac{\sqrt{2}}{2}}{- \frac{\sqrt{2}}{2}}$

$\rightarrow \tan \left({135}^{\circ}\right) = \frac{2}{\sqrt{2}} \cdot - \frac{2}{\sqrt{2}}$

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))

$\rightarrow \tan \left({150}^{\circ}\right) = \frac{\sin \left({150}^{\circ}\right)}{\cos \left({150}^{\circ}\right)}$

$\rightarrow \tan \left({150}^{\circ}\right) = \frac{\frac{1}{2}}{- \frac{\sqrt{3}}{2}}$

$\rightarrow \tan \left({150}^{\circ}\right) = \frac{1}{2} \cdot - \frac{2}{\sqrt{3}}$

$\rightarrow \tan \left({150}^{\circ}\right) = - \frac{\cancel{2}}{\cancel{2} \sqrt{3}}$

$\rightarrow \tan \left({150}^{\circ}\right) = - \frac{1}{3}$

Here is the most important part. The denominator is an irrational number, so multiplu both numerator and denominator by $\sqrt{3}$

$\rightarrow \tan \left({150}^{\circ}\right) = - \frac{1}{3} \cdot \left(\frac{\sqrt{3}}{\sqrt{3}}\right)$

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

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Hope this helps!!! :)