Trigonometry Examples

$\sin (θ) = \frac{2}{3}$ , $\cos (θ) = \frac{\sqrt{5}}{3}$

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.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\frac{2}{3}}{\frac{\sqrt{5}}{3}}$

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{2}{3} \cdot \frac{3}{\sqrt{5}}$

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Cancel the common factor.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{2}{3} \cdot \frac{3}{\sqrt{5}}$

Step 2.2.2

Rewrite the expression.

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

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

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

Step 2.5.2

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 2.5.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 2.5.4

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

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

Step 2.5.5

Add $1$ and $1$ .

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

Step 2.5.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

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

Step 2.5.6.2

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

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

Step 2.5.6.3

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

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

Step 2.5.6.4

Cancel the common factor of $2$ .

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Step 2.5.6.4.1

Cancel the common factor.

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

Step 2.5.6.4.2

Rewrite the expression.

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

Step 2.5.6.5

Evaluate the exponent.

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

Step 3

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

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

Step 4

Simplify the result.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Multiply $\frac{5}{2 \sqrt{5}}$ by $1$ .

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

Step 4.3

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

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

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

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

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

Step 4.4.2

Move $\sqrt{5}$ .

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

Step 4.4.3

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 4.4.4

Raise $\sqrt{5}$ to the power of $1$ .

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

Step 4.4.5

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

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{5 \sqrt{5}}{2 {\sqrt{5}}^{1 + 1}}$

Step 4.4.6

Add $1$ and $1$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{5 \sqrt{5}}{2 {\sqrt{5}}^{2}}$

Step 4.4.7

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{5 \sqrt{5}}{2 {(5^{\frac{1}{2}})}^{2}}$

Step 4.4.7.2

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

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

Step 4.4.7.3

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

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

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.

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

Step 4.4.7.4.2

Rewrite the expression.

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

Step 4.4.7.5

Evaluate the exponent.

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

Step 4.5

Cancel the common factor of $5$ .

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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

Step 5

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{1}{\frac{\sqrt{5}}{3}}$

Step 6

Simplify the result.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (θ) = \frac{1}{\cos (θ)} = 1 (\frac{3}{\sqrt{5}})$

Step 6.2

Multiply $\frac{3}{\sqrt{5}}$ by $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3}{\sqrt{5}}$

Step 6.3

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{3}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.4.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.4.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 6.4.4

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 6.4.5

Add $1$ and $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 6.4.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 6.4.6.2

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 6.4.6.3

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.4.6.4

Cancel the common factor of $2$ .

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Step 6.4.6.4.1

Cancel the common factor.

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 6.4.6.4.2

Rewrite the expression.

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{5}$

Step 6.4.6.5

Evaluate the exponent.

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{3 \sqrt{5}}{5}$

Step 7

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

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

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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

Step 9

The trig functions found are as follows:

$\sin (θ) = \frac{2}{3}$

$\cos (θ) = \frac{\sqrt{5}}{3}$

$\tan (θ) = \frac{2 \sqrt{5}}{5}$

$\cot (θ) = \frac{\sqrt{5}}{2}$

$\sec (θ) = \frac{3 \sqrt{5}}{5}$

$\csc (θ) = \frac{3}{2}$