Trigonometry Examples

$\arctan (\cot (\frac{2 π}{3}))$

Step 1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cotangent is negative in the second quadrant.

$\arctan (- \cot (\frac{π}{3}))$

Step 2

The exact value of $\cot (\frac{π}{3})$ is $\frac{1}{\sqrt{3}}$ .

$\arctan (- \frac{1}{\sqrt{3}})$

Step 3

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\arctan (- (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}))$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\arctan (- \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}})$

Step 4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}})$

Step 4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}})$

Step 4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}})$

Step 4.5

Add $1$ and $1$ .

$\arctan (- \frac{\sqrt{3}}{{\sqrt{3}}^{2}})$

Step 4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\arctan (- \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}})$

Step 4.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\arctan (- \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}})$

Step 4.6.3

Combine $\frac{1}{2}$ and $2$ .

$\arctan (- \frac{\sqrt{3}}{3^{\frac{2}{2}}})$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\arctan (- \frac{\sqrt{3}}{3^{\frac{2}{2}}})$

Step 4.6.4.2

Rewrite the expression.

$\arctan (- \frac{\sqrt{3}}{3^{1}})$

Step 4.6.5

Evaluate the exponent.

$\arctan (- \frac{\sqrt{3}}{3})$

Step 5

The exact value of $\arctan (- \frac{\sqrt{3}}{3})$ is $- \frac{π}{6}$ .

$- \frac{π}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{π}{6}$

Decimal Form:

$- 0.52359877 \dots$