Trigonometry Examples

$\sin (x) = \frac{2 \sqrt{2}}{3}$ , $\cos (x) = \frac{1}{3}$

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.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{\frac{2 \sqrt{2}}{3}}{\frac{1}{3}}$

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{2 \sqrt{2}}{3} \cdot 3$

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Cancel the common factor.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{2 \sqrt{2}}{3} \cdot 3$

Step 2.2.2

Rewrite the expression.

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

Step 3

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

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

Step 4

Simplify the result.

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Step 4.1

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

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

Step 4.2

Combine and simplify the denominator.

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Step 4.2.1

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

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

Step 4.2.2

Move $\sqrt{2}$ .

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 4.2.6

Add $1$ and $1$ .

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

Step 4.2.7

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

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Step 4.2.7.1

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

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

Step 4.2.7.2

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

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

Step 4.2.7.3

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

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

Step 4.2.7.4

Cancel the common factor of $2$ .

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Step 4.2.7.4.1

Cancel the common factor.

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

Step 4.2.7.4.2

Rewrite the expression.

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

Step 4.2.7.5

Evaluate the exponent.

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

Step 4.3

Multiply $2$ by $2$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{\sqrt{2}}{4}$

Step 5

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{1}{\frac{1}{3}}$

Step 6

Simplify the result.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (x) = \frac{1}{\cos (x)} = 1 \cdot 3$

Step 6.2

Multiply $3$ by $1$ .

$\sec (x) = \frac{1}{\cos (x)} = 3$

Step 7

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{1}{\frac{2 \sqrt{2}}{3}}$

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (x) = \frac{1}{\sin (x)} = 1 (\frac{3}{2 \sqrt{2}})$

Step 8.2

Multiply $\frac{3}{2 \sqrt{2}}$ by $1$ .

$\csc (x) = \frac{1}{\sin (x)} = \frac{3}{2 \sqrt{2}}$

Step 8.3

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 8.4

Combine and simplify the denominator.

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Step 8.4.1

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Step 8.4.2

Move $\sqrt{2}$ .

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 8.4.3

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 8.4.4

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Step 8.4.5

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Step 8.4.6

Add $1$ and $1$ .

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

Step 8.4.7

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

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Step 8.4.7.1

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

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

Step 8.4.7.2

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 8.4.7.3

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

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 8.4.7.4

Cancel the common factor of $2$ .

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Step 8.4.7.4.1

Cancel the common factor.

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Step 8.4.7.4.2

Rewrite the expression.

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 8.4.7.5

Evaluate the exponent.

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{2 \cdot 2}$

Step 8.5

Multiply $2$ by $2$ .

$\csc (x) = \frac{1}{\sin (x)} = \frac{3 \sqrt{2}}{4}$

Step 9

The trig functions found are as follows:

$\sin (x) = \frac{2 \sqrt{2}}{3}$

$\cos (x) = \frac{1}{3}$

$\tan (x) = 2 \sqrt{2}$

$\cot (x) = \frac{\sqrt{2}}{4}$

$\sec (x) = 3$

$\csc (x) = \frac{3 \sqrt{2}}{4}$