Question

Evaluate cot 120° without using a calculator by using ratios in a reference triangle.

Answer 1

The reference angle for $120\degree$ is  $60\degree$.

Recall that  $cot(120\degree) = \frac{1}{tan(120\degree)}$

This implies that  $cot(120\degree) = \frac{1}{tan(60\degree)}$

Recall also from $30\degree - 60\degreee - 90\degree$ triangle that

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

Putting all together we have

$cot(120\degree) = \frac{1}{-\sqrt(3)}$

Rationalizing the denominator gives

$cot(120\degree) = \frac{-\sqrt(3)}{3}$

Answer 2

Answer:

The value of cot 120° is $-\frac{\sqrt{3}}{3}$.

Step-by-step explanation:

Consider the trigonometric identity:

$cot(180-\theta)= -cot(\theta)$

Now use the above trigonometric identity:

$cot(120)= cot(180-60)$

$cot(180-60)= -cot(60)$

Now use the identity: $cot(\theta)=\frac{1}{tan(\theta)}$

$-cot(60)=-\frac{1}{tan(60)}$

Substitute the value of tan 60°.

$-\frac{1}{tan(60)}=-\frac{1}{\sqrt{3}}$

Now rationalize the denominator gives us:

$-\frac{\sqrt{3}}{3}$

Hence, the value of cot 120° is $-\frac{\sqrt{3}}{3}$.