Precalculus Examples

$\sin (θ) = \frac{1}{4}$ , $\cos (θ) = \frac{\sqrt{15}}{4}$

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.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\frac{1}{4}}{\frac{\sqrt{15}}{4}}$

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{1}{4} \cdot \frac{4}{\sqrt{15}}$

Step 2.2

Cancel the common factor of $4$ .

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Step 2.2.1

Cancel the common factor.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{1}{4} \cdot \frac{4}{\sqrt{15}}$

Step 2.2.2

Rewrite the expression.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{1}{\sqrt{15}}$

Step 2.3

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{1}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}$

Step 2.4

Combine and simplify the denominator.

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Step 2.4.1

Multiply $\frac{1}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 2.4.2

Raise $\sqrt{15}$ to the power of $1$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 2.4.3

Raise $\sqrt{15}$ to the power of $1$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 2.4.4

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

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Step 2.4.5

Add $1$ and $1$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{{\sqrt{15}}^{2}}$

Step 2.4.6

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 2.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

Step 2.4.6.2

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

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

Step 2.4.6.3

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

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{15^{\frac{2}{2}}}$

Step 2.4.6.4

Cancel the common factor of $2$ .

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Step 2.4.6.4.1

Cancel the common factor.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{15^{\frac{2}{2}}}$

Step 2.4.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{15}$

Step 2.4.6.5

Evaluate the exponent.

$\tan (θ) = \frac{\sin (θ)}{\cos (θ)} = \frac{\sqrt{15}}{15}$

Step 3

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

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{1}{\frac{\sqrt{15}}{15}}$

Step 4

Simplify the result.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (θ) = \frac{1}{\tan (θ)} = 1 (\frac{15}{\sqrt{15}})$

Step 4.2

Multiply $\frac{15}{\sqrt{15}}$ by $1$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15}{\sqrt{15}}$

Step 4.3

Multiply $\frac{15}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{15}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 4.4.2

Raise $\sqrt{15}$ to the power of $1$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 4.4.3

Raise $\sqrt{15}$ to the power of $1$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 4.4.4

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

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Step 4.4.5

Add $1$ and $1$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{{\sqrt{15}}^{2}}$

Step 4.4.6

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 4.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

Step 4.4.6.2

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

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

Step 4.4.6.3

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

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{15^{\frac{2}{2}}}$

Step 4.4.6.4

Cancel the common factor of $2$ .

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Step 4.4.6.4.1

Cancel the common factor.

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{15^{\frac{2}{2}}}$

Step 4.4.6.4.2

Rewrite the expression.

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{15}$

Step 4.4.6.5

Evaluate the exponent.

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{15}$

Step 4.5

Cancel the common factor of $15$ .

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Step 4.5.1

Cancel the common factor.

$\cot (θ) = \frac{1}{\tan (θ)} = \frac{15 \sqrt{15}}{15}$

Step 4.5.2

Divide $\sqrt{15}$ by $1$ .

$\cot (θ) = \frac{1}{\tan (θ)} = \sqrt{15}$

Step 5

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{1}{\frac{\sqrt{15}}{4}}$

Step 6

Simplify the result.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (θ) = \frac{1}{\cos (θ)} = 1 (\frac{4}{\sqrt{15}})$

Step 6.2

Multiply $\frac{4}{\sqrt{15}}$ by $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4}{\sqrt{15}}$

Step 6.3

Multiply $\frac{4}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4}{\sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{4}{\sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 6.4.2

Raise $\sqrt{15}$ to the power of $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 6.4.3

Raise $\sqrt{15}$ to the power of $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{\sqrt{15} \sqrt{15}}$

Step 6.4.4

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{{\sqrt{15}}^{1 + 1}}$

Step 6.4.5

Add $1$ and $1$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{{\sqrt{15}}^{2}}$

Step 6.4.6

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{{(15^{\frac{1}{2}})}^{2}}$

Step 6.4.6.2

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{15^{\frac{1}{2} \cdot 2}}$

Step 6.4.6.3

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

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{15^{\frac{2}{2}}}$

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.

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{15^{\frac{2}{2}}}$

Step 6.4.6.4.2

Rewrite the expression.

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{15}$

Step 6.4.6.5

Evaluate the exponent.

$\sec (θ) = \frac{1}{\cos (θ)} = \frac{4 \sqrt{15}}{15}$

Step 7

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

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

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

$\csc (θ) = \frac{1}{\sin (θ)} = 1 \cdot 4$

Step 8.2

Multiply $4$ by $1$ .

$\csc (θ) = \frac{1}{\sin (θ)} = 4$

Step 9

The trig functions found are as follows:

$\sin (θ) = \frac{1}{4}$

$\cos (θ) = \frac{\sqrt{15}}{4}$

$\tan (θ) = \frac{\sqrt{15}}{15}$

$\cot (θ) = \sqrt{15}$

$\sec (θ) = \frac{4 \sqrt{15}}{15}$

$\csc (θ) = 4$