Trigonometry Examples

$\cot (60) \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cot (60)$ is $\frac{1}{\sqrt{3}}$ .

$\frac{1}{\sqrt{3}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.2

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

$\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

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

$\frac{\sqrt{3}}{\sqrt{3} \sqrt{3}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.2

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.3

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.4

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

$\frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.5

Add $1$ and $1$ .

$\frac{\sqrt{3}}{{\sqrt{3}}^{2}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.6

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

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Step 1.3.6.1

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

$\frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.6.2

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

$\frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.6.3

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

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\frac{\sqrt{3}}{3^{\frac{2}{2}}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.6.4.2

Rewrite the expression.

$\frac{\sqrt{3}}{3^{1}} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.3.6.5

Evaluate the exponent.

$\frac{\sqrt{3}}{3} \cdot (2 \tan (30)) + \sec^{2} (45)$

Step 1.4

The exact value of $\tan (30)$ is $\frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3} \cdot (2 \frac{\sqrt{3}}{3}) + \sec^{2} (45)$

Step 1.5

Combine $2$ and $\frac{\sqrt{3}}{3}$ .

$\frac{\sqrt{3}}{3} \cdot \frac{2 \sqrt{3}}{3} + \sec^{2} (45)$

Step 1.6

Multiply $\frac{\sqrt{3}}{3} \cdot \frac{2 \sqrt{3}}{3}$ .

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Step 1.6.1

Multiply $\frac{\sqrt{3}}{3}$ by $\frac{2 \sqrt{3}}{3}$ .

$\frac{\sqrt{3} (2 \sqrt{3})}{3 \cdot 3} + \sec^{2} (45)$

Step 1.6.2

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

$\frac{2 ({\sqrt{3}}^{1} \sqrt{3})}{3 \cdot 3} + \sec^{2} (45)$

Step 1.6.3

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

$\frac{2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}{3 \cdot 3} + \sec^{2} (45)$

Step 1.6.4

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

$\frac{2 {\sqrt{3}}^{1 + 1}}{3 \cdot 3} + \sec^{2} (45)$

Step 1.6.5

Add $1$ and $1$ .

$\frac{2 {\sqrt{3}}^{2}}{3 \cdot 3} + \sec^{2} (45)$

Step 1.6.6

Multiply $3$ by $3$ .

$\frac{2 {\sqrt{3}}^{2}}{9} + \sec^{2} (45)$

Step 1.7

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

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Step 1.7.1

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

$\frac{2 {(3^{\frac{1}{2}})}^{2}}{9} + \sec^{2} (45)$

Step 1.7.2

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

$\frac{2 \cdot 3^{\frac{1}{2} \cdot 2}}{9} + \sec^{2} (45)$

Step 1.7.3

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

$\frac{2 \cdot 3^{\frac{2}{2}}}{9} + \sec^{2} (45)$

Step 1.7.4

Cancel the common factor of $2$ .

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Step 1.7.4.1

Cancel the common factor.

$\frac{2 \cdot 3^{\frac{2}{2}}}{9} + \sec^{2} (45)$

Step 1.7.4.2

Rewrite the expression.

$\frac{2 \cdot 3^{1}}{9} + \sec^{2} (45)$

Step 1.7.5

Evaluate the exponent.

$\frac{2 \cdot 3}{9} + \sec^{2} (45)$

Step 1.8

Multiply $2$ by $3$ .

$\frac{6}{9} + \sec^{2} (45)$

Step 1.9

Cancel the common factor of $6$ and $9$ .

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Step 1.9.1

Factor $3$ out of $6$ .

$\frac{3 (2)}{9} + \sec^{2} (45)$

Step 1.9.2

Cancel the common factors.

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Step 1.9.2.1

Factor $3$ out of $9$ .

$\frac{3 \cdot 2}{3 \cdot 3} + \sec^{2} (45)$

Step 1.9.2.2

Cancel the common factor.

$\frac{3 \cdot 2}{3 \cdot 3} + \sec^{2} (45)$

Step 1.9.2.3

Rewrite the expression.

$\frac{2}{3} + \sec^{2} (45)$

Step 1.10

The exact value of $\sec (45)$ is $\frac{2}{\sqrt{2}}$ .

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

Step 1.11

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\frac{2}{3} + {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2}$

Step 1.12

Combine and simplify the denominator.

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Step 1.12.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 1.12.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.12.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 1.12.4

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

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

Step 1.12.5

Add $1$ and $1$ .

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

Step 1.12.6

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

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Step 1.12.6.1

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

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

Step 1.12.6.2

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

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

Step 1.12.6.3

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

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

Step 1.12.6.4

Cancel the common factor of $2$ .

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Step 1.12.6.4.1

Cancel the common factor.

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

Step 1.12.6.4.2

Rewrite the expression.

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

Step 1.12.6.5

Evaluate the exponent.

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

Step 1.13

Cancel the common factor of $2$ .

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Step 1.13.1

Cancel the common factor.

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

Step 1.13.2

Divide $\sqrt{2}$ by $1$ .

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

Step 1.14

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

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Step 1.14.1

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

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

Step 1.14.2

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

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

Step 1.14.3

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

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

Step 1.14.4

Cancel the common factor of $2$ .

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Step 1.14.4.1

Cancel the common factor.

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

Step 1.14.4.2

Rewrite the expression.

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

Step 1.14.5

Evaluate the exponent.

$\frac{2}{3} + 2$

Step 2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2}{3} + 2 \cdot \frac{3}{3}$

Step 3

Combine $2$ and $\frac{3}{3}$ .

$\frac{2}{3} + \frac{2 \cdot 3}{3}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 + 2 \cdot 3}{3}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply $2$ by $3$ .

$\frac{2 + 6}{3}$

Step 5.2

Add $2$ and $6$ .

$\frac{8}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{8}{3}$

Decimal Form:

$2.\overline{6}$

Mixed Number Form:

$2 \frac{2}{3}$