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The reference angle for [tex] 120\degree[/tex] is  [tex] 60\degree[/tex].

Recall that  [tex] cot(120\degree) = \frac{1}{tan(120\degree)}[/tex]

This implies that  [tex] cot(120\degree) = \frac{1}{tan(60\degree)}[/tex]

Recall also from [tex] 30\degree - 60\degreee - 90\degree[/tex] triangle that

[tex] tan(60\degree)=\sqrt(3)[/tex] and also since [tex] 120\degree [/tex] is the in the second quadrant, the tangent ratio is negative.

Putting all together we have

[tex] cot(120\degree) = \frac{1}{-\sqrt(3)}[/tex]

Rationalizing the denominator gives

[tex] cot(120\degree) = \frac{-\sqrt(3)}{3}[/tex]

Answer:

The value of cot 120° is [tex]-\frac{\sqrt{3}}{3}[/tex].

Step-by-step explanation:

Consider the trigonometric identity:

[tex]cot(180-\theta)= -cot(\theta)[/tex]

Now use the above trigonometric identity:

[tex]cot(120)= cot(180-60)[/tex]

[tex]cot(180-60)= -cot(60)[/tex]

Now use the identity: [tex]cot(\theta)=\frac{1}{tan(\theta)}[/tex]

[tex]-cot(60)=-\frac{1}{tan(60)}[/tex]

Substitute the value of tan 60°.

[tex]-\frac{1}{tan(60)}=-\frac{1}{\sqrt{3}}[/tex]

Now rationalize the denominator gives us:

[tex]-\frac{\sqrt{3}}{3}[/tex]

Hence, the value of cot 120° is [tex]-\frac{\sqrt{3}}{3}[/tex].

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