Trigonometry Examples

$\sin (x) = \frac{5}{7}$ , $\cos (x) = \frac{2 \sqrt{6}}{7}$

Step 1

To find the value of $\tan (x)$ , use the fact that $\tan (x) = \frac{\sin (x)}{\cos (x)}$ then substitute in the known values.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{\frac{5}{7}}{\frac{2 \sqrt{6}}{7}}$

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5}{7} \cdot \frac{7}{2 \sqrt{6}}$

Step 2.2

Cancel the common factor of $7$ .

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Step 2.2.1

Cancel the common factor.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5}{7} \cdot \frac{7}{2 \sqrt{6}}$

Step 2.2.2

Rewrite the expression.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = 5 (\frac{1}{2 \sqrt{6}})$

Step 2.3

Combine $5$ and $\frac{1}{2 \sqrt{6}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5}{2 \sqrt{6}}$

Step 2.4

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 2.5.2

Move $\sqrt{6}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 2.5.3

Raise $\sqrt{6}$ to the power of $1$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 2.5.4

Raise $\sqrt{6}$ to the power of $1$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 2.5.5

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

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 2.5.6

Add $1$ and $1$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 2.5.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 2.5.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Step 2.5.7.2

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

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 2.5.7.3

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

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 2.5.7.4

Cancel the common factor of $2$ .

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Step 2.5.7.4.1

Cancel the common factor.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 2.5.7.4.2

Rewrite the expression.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 \cdot 6}$

Step 2.5.7.5

Evaluate the exponent.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{2 \cdot 6}$

Step 2.6

Multiply $2$ by $6$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{5 \sqrt{6}}{12}$

Step 3

To find the value of $\cot (x)$ , use the fact that $\frac{1}{\tan (x)}$ then substitute in the known values.

$\cot (x) = \frac{1}{\tan (x)} = \frac{1}{\frac{5 \sqrt{6}}{12}}$

Step 4

Simplify the result.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (x) = \frac{1}{\tan (x)} = 1 (\frac{12}{5 \sqrt{6}})$

Step 4.2

Multiply $\frac{12}{5 \sqrt{6}}$ by $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12}{5 \sqrt{6}}$

Step 4.3

Multiply $\frac{12}{5 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12}{5 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{12}{5 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 \sqrt{6} \sqrt{6}}$

Step 4.4.2

Move $\sqrt{6}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 (\sqrt{6} \sqrt{6})}$

Step 4.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 (\sqrt{6} \sqrt{6})}$

Step 4.4.4

Raise $\sqrt{6}$ to the power of $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 (\sqrt{6} \sqrt{6})}$

Step 4.4.5

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 {\sqrt{6}}^{1 + 1}}$

Step 4.4.6

Add $1$ and $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 {\sqrt{6}}^{2}}$

Step 4.4.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 4.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 {(6^{\frac{1}{2}})}^{2}}$

Step 4.4.7.2

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 4.4.7.3

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 \cdot 6^{\frac{2}{2}}}$

Step 4.4.7.4

Cancel the common factor of $2$ .

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Step 4.4.7.4.1

Cancel the common factor.

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 \cdot 6^{\frac{2}{2}}}$

Step 4.4.7.4.2

Rewrite the expression.

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 \cdot 6}$

Step 4.4.7.5

Evaluate the exponent.

$\cot (x) = \frac{1}{\tan (x)} = \frac{12 \sqrt{6}}{5 \cdot 6}$

Step 4.5

Cancel the common factor of $12$ and $6$ .

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Step 4.5.1

Factor $6$ out of $12 \sqrt{6}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{6 (2 \sqrt{6})}{5 \cdot 6}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $6$ out of $5 \cdot 6$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{6 (2 \sqrt{6})}{6 \cdot 5}$

Step 4.5.2.2

Cancel the common factor.

$\cot (x) = \frac{1}{\tan (x)} = \frac{6 (2 \sqrt{6})}{6 \cdot 5}$

Step 4.5.2.3

Rewrite the expression.

$\cot (x) = \frac{1}{\tan (x)} = \frac{2 \sqrt{6}}{5}$

Step 5

To find the value of $\sec (x)$ , use the fact that $\frac{1}{\cos (x)}$ then substitute in the known values.

$\sec (x) = \frac{1}{\cos (x)} = \frac{1}{\frac{2 \sqrt{6}}{7}}$

Step 6

Simplify the result.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (x) = \frac{1}{\cos (x)} = 1 (\frac{7}{2 \sqrt{6}})$

Step 6.2

Multiply $\frac{7}{2 \sqrt{6}}$ by $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7}{2 \sqrt{6}}$

Step 6.3

Multiply $\frac{7}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{7}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 6.4.2

Move $\sqrt{6}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 6.4.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 6.4.4

Raise $\sqrt{6}$ to the power of $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 6.4.5

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 6.4.6

Add $1$ and $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 {\sqrt{6}}^{2}}$

Step 6.4.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 6.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 {(6^{\frac{1}{2}})}^{2}}$

Step 6.4.7.2

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 \cdot 6^{\frac{1}{2} \cdot 2}}$

Step 6.4.7.3

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 6.4.7.4

Cancel the common factor of $2$ .

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Step 6.4.7.4.1

Cancel the common factor.

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 6.4.7.4.2

Rewrite the expression.

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 \cdot 6}$

Step 6.4.7.5

Evaluate the exponent.

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{2 \cdot 6}$

Step 6.5

Multiply $2$ by $6$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{7 \sqrt{6}}{12}$

Step 7

To find the value of $\csc (x)$ , use the fact that $\frac{1}{\sin (x)}$ then substitute in the known values.

$\csc (x) = \frac{1}{\sin (x)} = \frac{1}{\frac{5}{7}}$

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (x) = \frac{1}{\sin (x)} = 1 (\frac{7}{5})$

Step 8.2

Multiply $\frac{7}{5}$ by $1$ .

$\csc (x) = \frac{1}{\sin (x)} = \frac{7}{5}$

Step 9

The trig functions found are as follows:

$\sin (x) = \frac{5}{7}$

$\cos (x) = \frac{2 \sqrt{6}}{7}$

$\tan (x) = \frac{5 \sqrt{6}}{12}$

$\cot (x) = \frac{2 \sqrt{6}}{5}$

$\sec (x) = \frac{7 \sqrt{6}}{12}$

$\csc (x) = \frac{7}{5}$