Trigonometry Examples

cot (- 300)

$\cot (- 300)$

Step 1

Rewrite

- 300

$- 300$ as an angle where the values of the six trigonometric functions are known divided by

2

$2$ .

cot (\frac{- 600}{2})

$\cot (\frac{- 600}{2})$

Step 2

Apply the reciprocal identity.

\frac{1}{tan (\frac{- 600}{2})}

$\frac{1}{\tan (\frac{- 600}{2})}$

Step 3

Apply the tangent half-angle identity.

\frac{1}{\pm \sqrt{\frac{1 - cos (- 600)}{1 + cos (- 600)}}}

$\frac{1}{\pm \sqrt{\frac{1 - \cos (- 600)}{1 + \cos (- 600)}}}$

Step 4

Change the

\pm

$\pm$ to

+

$+$ because cotangent is positive in the first quadrant.

\frac{1}{\sqrt{\frac{1 - cos (- 600)}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 - \cos (- 600)}{1 + \cos (- 600)}}}$

Step 5

Simplify

\frac{1}{\sqrt{\frac{1 - cos (- 600)}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 - \cos (- 600)}{1 + \cos (- 600)}}}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Add full rotations of

360

$360$ ° until the angle is between

0

$0$ ° and

360

$360$ °.

\frac{1}{\sqrt{\frac{1 - cos (120)}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 - \cos (120)}{1 + \cos (- 600)}}}$

Step 5.1.2

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

\frac{1}{\sqrt{\frac{1 - - cos (60)}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 - - \cos (60)}{1 + \cos (- 600)}}}$

Step 5.1.3

The exact value of

cos (60)

$\cos (60)$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{\sqrt{\frac{1 - - \frac{1}{2}}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 - - \frac{1}{2}}{1 + \cos (- 600)}}}$

Step 5.1.4

Multiply

- - \frac{1}{2}

$- - \frac{1}{2}$ .

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Step 5.1.4.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\frac{1}{\sqrt{\frac{1 + 1 (\frac{1}{2})}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 + 1 (\frac{1}{2})}{1 + \cos (- 600)}}}$

Step 5.1.4.2

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

1

$1$ .

\frac{1}{\sqrt{\frac{1 + \frac{1}{2}}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 + \frac{1}{2}}{1 + \cos (- 600)}}}$

\frac{1}{\sqrt{\frac{1 + \frac{1}{2}}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{1 + \frac{1}{2}}{1 + \cos (- 600)}}}$

Step 5.1.5

Write

1

$1$ as a fraction with a common denominator.

\frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{1}{2}}{1 + cos (- 600)}}}

$\frac{1}{\sqrt{\frac{\frac{2}{2} + \frac{1}{2}}{1 + \cos (- 600)}}}$

Step 5.1.6

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{\frac{2 + 1}{2}}{1 + \cos (- 600)}}}$

Step 5.1.7

Add

$2$ and

$1$ .

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{1 + \cos (- 600)}}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Add full rotations of

$360$ ° until the angle is between

$0$ ° and

$360$ °.

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{1 + \cos (120)}}}$

Step 5.2.2

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

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{1 - \cos (60)}}}$

Step 5.2.3

The exact value of

$\cos (60)$ is

$\frac{1}{2}$ .

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{1 - \frac{1}{2}}}}$

Step 5.2.4

Write

$1$ as a fraction with a common denominator.

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{\frac{2}{2} - \frac{1}{2}}}}$

Step 5.2.5

Combine the numerators over the common denominator.

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{\frac{2 - 1}{2}}}}$

Step 5.2.6

Subtract

$1$ from

$2$ .

$\frac{1}{\sqrt{\frac{\frac{3}{2}}{\frac{1}{2}}}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\sqrt{\frac{3}{2} \cdot 2}}$

Step 5.3.2

Cancel the common factor of

$2$ .

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Step 5.3.2.1

Cancel the common factor.

$\frac{1}{\sqrt{\frac{3}{2} \cdot 2}}$

Step 5.3.2.2

Rewrite the expression.

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

Step 5.4

Multiply

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

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

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

Step 5.5

Combine and simplify the denominator.

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Step 5.5.1

Multiply

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

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

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

Step 5.5.2

Raise

$\sqrt{3}$ to the power of

$1$ .

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

Step 5.5.3

Raise

$\sqrt{3}$ to the power of

$1$ .

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

Step 5.5.4

Use the power rule

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.5.5

Add

$1$ and

$1$ .

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

Step 5.5.6

Rewrite

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

$3$ .

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Step 5.5.6.1

Use

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

$\sqrt{3}$ as

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

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

Step 5.5.6.2

Apply the power rule and multiply exponents,

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

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.5.6.3

Combine

$\frac{1}{2}$ and

$2$ .

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

Step 5.5.6.4

Cancel the common factor of

$2$ .

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Step 5.5.6.4.1

Cancel the common factor.

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

Step 5.5.6.4.2

Rewrite the expression.

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

Step 5.5.6.5

Evaluate the exponent.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.57735026 \dots$