Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the result.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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

Step 3.3

Combine and simplify the denominator.

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Step 3.3.1

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

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

Step 3.3.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.3.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 3.3.4

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

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

Step 3.3.5

Add $1$ and $1$ .

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

Step 3.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3.3.6.2

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

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

Step 3.3.6.3

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

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

Step 3.3.6.4

Cancel the common factor of $2$ .

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Step 3.3.6.4.1

Cancel the common factor.

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

Step 3.3.6.4.2

Rewrite the expression.

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

Step 3.3.6.5

Evaluate the exponent.

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

Step 3.4

Cancel the common factor of $3$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Divide $\sqrt{3}$ by $1$ .

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

Step 3.5

Combine $\frac{1}{2}$ and $\sqrt{3}$ .

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

Step 4

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{1}{\frac{\sqrt{3}}{3}}$

Step 5

Simplify the result.

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Step 5.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (x) = \frac{1}{\tan (x)} = 1 (\frac{3}{\sqrt{3}})$

Step 5.2

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{3}{\sqrt{3}}$

Step 5.3

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{3}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.4

Combine and simplify the denominator.

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Step 5.4.1

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.4.2

Raise $\sqrt{3}$ to the power of $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.4.3

Raise $\sqrt{3}$ to the power of $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.4.4

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.4.5

Add $1$ and $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.4.6.2

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.4.6.3

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.4.6.4

Cancel the common factor of $2$ .

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Step 5.4.6.4.1

Cancel the common factor.

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.4.6.4.2

Rewrite the expression.

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{3}$

Step 5.4.6.5

Evaluate the exponent.

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{3}$

Step 5.5

Cancel the common factor of $3$ .

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Step 5.5.1

Cancel the common factor.

$\cot (x) = \frac{1}{\tan (x)} = \frac{3 \sqrt{3}}{3}$

Step 5.5.2

Divide $\sqrt{3}$ by $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \sqrt{3}$

Step 6

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

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

Step 7

Simplify the result.

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Step 7.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 7.2

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

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

Step 7.3

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

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

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

Step 7.4.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 7.4.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 7.4.4

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.4.5

Add $1$ and $1$ .

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

Step 7.4.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 7.4.6.2

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

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

Step 7.4.6.3

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

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

Step 7.4.6.4

Cancel the common factor of $2$ .

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Step 7.4.6.4.1

Cancel the common factor.

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

Step 7.4.6.4.2

Rewrite the expression.

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

Step 7.4.6.5

Evaluate the exponent.

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

Step 8

The trig functions found are as follows:

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

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

$\tan (x) = \frac{\sqrt{3}}{3}$

$\cot (x) = \sqrt{3}$

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

$\csc (x) = 2$