Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{2 \sqrt{2}}{3} \cdot (- 1 \cdot 3)$

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Factor $3$ out of $- 1 \cdot 3$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{2 \sqrt{2}}{3} \cdot (3 \cdot - 1)$

Step 2.2.2

Cancel the common factor.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{2 \sqrt{2}}{3} \cdot (3 \cdot - 1)$

Step 2.2.3

Rewrite the expression.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = 2 \sqrt{2} \cdot - 1$

Step 2.3

Multiply $- 1$ by $2$ .

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

Step 3

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

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

Step 4

Simplify the result.

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Step 4.1

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

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

Step 4.3.2

Move $\sqrt{2}$ .

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

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

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

Step 4.3.6

Add $1$ and $1$ .

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

Step 4.3.7

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

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Step 4.3.7.1

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

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

Step 4.3.7.2

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

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

Step 4.3.7.3

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

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

Step 4.3.7.4

Cancel the common factor of $2$ .

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Step 4.3.7.4.1

Cancel the common factor.

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

Step 4.3.7.4.2

Rewrite the expression.

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

Step 4.3.7.5

Evaluate the exponent.

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

Step 4.4

Multiply $2$ by $2$ .

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

Step 5

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

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

Step 6

Simplify the result.

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Step 6.1

Cancel the common factor of $1$ and $- 1$ .

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Step 6.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 6.1.2

Move the negative in front of the fraction.

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

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3

Multiply $- (1 \cdot 3)$ .

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Step 6.3.1

Multiply $3$ by $1$ .

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

Step 6.3.2

Multiply $- 1$ by $3$ .

$\sec (θ) = \frac{1}{\cos (θ)} = - 3$

Step 7

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

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

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Combine and simplify the denominator.

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Step 8.4.1

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

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

Step 8.4.2

Move $\sqrt{2}$ .

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

Step 8.4.3

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

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

Step 8.4.4

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

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

Step 8.4.5

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

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

Step 8.4.6

Add $1$ and $1$ .

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

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

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

Step 8.4.7.2

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

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

Step 8.4.7.3

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

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

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.

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

Step 8.4.7.4.2

Rewrite the expression.

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

Step 8.4.7.5

Evaluate the exponent.

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

Step 8.5

Multiply $2$ by $2$ .

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

Step 9

The trig functions found are as follows:

$\sin (θ) = \frac{2 \sqrt{2}}{3}$

$\cos (θ) = - \frac{1}{3}$

$\tan (θ) = - 2 \sqrt{2}$

$\cot (θ) = - \frac{\sqrt{2}}{4}$

$\sec (θ) = - 3$

$\csc (θ) = \frac{3 \sqrt{2}}{4}$