Trigonometry Examples

$arccot (x) - \arccos (\frac{1}{2}) = \frac{π}{3}$

Step 1

Simplify the left side.

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Step 1.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$arccot (x) - \frac{π}{3} = \frac{π}{3}$

Step 2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.1

Add $\frac{π}{3}$ to both sides of the equation.

$arccot (x) = \frac{π}{3} + \frac{π}{3}$

Step 2.2

Combine the numerators over the common denominator.

$arccot (x) = \frac{π + π}{3}$

Step 2.3

Add $π$ and $π$ .

$arccot (x) = \frac{2 π}{3}$

Step 3

Take the inverse arccotangent of both sides of the equation to extract $x$ from inside the arccotangent.

$x = \cot (\frac{2 π}{3})$

Step 4

Simplify the right side.

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Step 4.1

Simplify $\cot (\frac{2 π}{3})$ .

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Step 4.1.1

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.

$x = - \cot (\frac{π}{3})$

Step 4.1.2

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

$x = - \frac{1}{\sqrt{3}}$

Step 4.1.3

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

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

Step 4.1.4

Combine and simplify the denominator.

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Step 4.1.4.1

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

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

Step 4.1.4.2

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

$x = - \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 4.1.4.3

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

$x = - \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 4.1.4.4

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

$x = - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 4.1.4.5

Add $1$ and $1$ .

$x = - \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 4.1.4.6

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

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Step 4.1.4.6.1

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

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

Step 4.1.4.6.2

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

$x = - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 4.1.4.6.3

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

$x = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.4.6.4

Cancel the common factor of $2$ .

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Step 4.1.4.6.4.1

Cancel the common factor.

$x = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 4.1.4.6.4.2

Rewrite the expression.

$x = - \frac{\sqrt{3}}{3^{1}}$

Step 4.1.4.6.5

Evaluate the exponent.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.57735026 \dots$