Trigonometry Examples

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

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

Step 1

Simplify the left side.

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Step 1.1

The exact value of

arccos (\frac{1}{2})

$\arccos (\frac{1}{2})$ is

\frac{π}{3}

$\frac{π}{3}$ .

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

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

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

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

Step 2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 2.1

Add

\frac{π}{3}

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

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

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

Step 2.2

Combine the numerators over the common denominator.

arccot (x) = \frac{π + π}{3}

$arccot (x) = \frac{π + π}{3}$

Step 2.3

Add

π

$π$ and

π

$π$ .

arccot (x) = \frac{2 π}{3}

$arccot (x) = \frac{2 π}{3}$

arccot (x) = \frac{2 π}{3}

$arccot (x) = \frac{2 π}{3}$

Step 3

Take the inverse arccotangent of both sides of the equation to extract

x

$x$ from inside the arccotangent.

x = cot (\frac{2 π}{3})

$x = \cot (\frac{2 π}{3})$

Step 4

Simplify the right side.

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Step 4.1

Simplify

cot (\frac{2 π}{3})

$\cot (\frac{2 π}{3})$ .

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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})

$x = - \cot (\frac{π}{3})$

Step 4.1.2

The exact value of

cot (\frac{π}{3})

$\cot (\frac{π}{3})$ is

\frac{1}{\sqrt{3}}

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

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

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

Step 4.1.3

Multiply

\frac{1}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

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

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

Step 4.1.4.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

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

Step 4.1.4.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

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

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 4.1.4.5

Add

1

$1$ and

1

$1$ .

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

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

Step 4.1.4.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 4.1.4.6.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 4.1.4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 4.1.4.6.4

Cancel the common factor of

2

$2$ .

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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:

$x = - 0.57735026 \dots$