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

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

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Answer 1 · 2017-03-13T21:51:38

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

Explanation:

Use $tan θ = \frac{sin (θ)}{cos (θ)}$ $tan theta = sin(theta)/cos(theta)$

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

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

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

Answer 2 · 2017-03-14T05:12:06

$\Rightarrow tan (120^{\circ}) = - \sqrt{3}$ $color(blue)(rArrtan(120^circ)=-sqrt(3)$

$\Rightarrow tan (135^{\circ}) = - 1$ $color(orange)(rArrtan(135^circ)=-1$

$\Rightarrow tan (150^{\circ}) = - \frac{\sqrt{3}}{3}$ $color(purple)(rArrtan(150^circ)=-(sqrt(3))/3$

Explanation:

Let's use the unit circle to find the values

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$tan (120^{\circ})$ $color(blue)(tan(120^circ)$

We have the values of $sin (120^{\circ}) and cos (120^{\circ})$ $sin(120^circ) and cos(120^circ)$

So, use the identity

$tan (θ) = \frac{sin (θ)}{cos (θ)}$ $color(brown)(tan(theta)=( sin(theta))/(cos(theta))$

$\to tan (120^{\circ}) = \frac{sin (120^{\circ})}{cos (120^{\circ})}$ $rarrtan(120^circ)=(sin(120^circ))/(cos(120^circ))$

$\to tan (120^{\circ}) = \frac{\frac{\sqrt{3}}{2}}{- \frac{1}{2}}$ $rarrtan(120^circ)=(sqrt(3)/2)/(-1/2)$

$\to tan (120^{\circ}) = \frac{\sqrt{3}}{2} \cdot - \frac{2}{1}$ $rarrtan(120^circ)=sqrt(3)/2*-2/1$

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

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

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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

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

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

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

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

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

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

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

enter image source here

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

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

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

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

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

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

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

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

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

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Hope this helps!!! :)

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

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