Trigonometry Examples

$\sin (x) = \frac{3}{4}$ , $\cos (x) = \frac{\sqrt{7}}{4}$

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.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{\frac{3}{4}}{\frac{\sqrt{7}}{4}}$

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3}{4} \cdot \frac{4}{\sqrt{7}}$

Step 2.2

Cancel the common factor of $4$ .

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Step 2.2.1

Cancel the common factor.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3}{4} \cdot \frac{4}{\sqrt{7}}$

Step 2.2.2

Rewrite the expression.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = 3 (\frac{1}{\sqrt{7}})$

Step 2.3

Combine $3$ and $\frac{1}{\sqrt{7}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3}{\sqrt{7}}$

Step 2.4

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 2.5

Combine and simplify the denominator.

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Step 2.5.1

Multiply $\frac{3}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.5.2

Raise $\sqrt{7}$ to the power of $1$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.5.3

Raise $\sqrt{7}$ to the power of $1$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 2.5.4

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

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 2.5.5

Add $1$ and $1$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 2.5.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 2.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 2.5.6.2

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

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 2.5.6.3

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

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

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.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 2.5.6.4.2

Rewrite the expression.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{7}$

Step 2.5.6.5

Evaluate the exponent.

$\tan (x) = \frac{\sin (x)}{\cos (x)} = \frac{3 \sqrt{7}}{7}$

Step 3

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{1}{\frac{3 \sqrt{7}}{7}}$

Step 4

Simplify the result.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\cot (x) = \frac{1}{\tan (x)} = 1 (\frac{7}{3 \sqrt{7}})$

Step 4.2

Multiply $\frac{7}{3 \sqrt{7}}$ by $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7}{3 \sqrt{7}}$

Step 4.3

Multiply $\frac{7}{3 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7}{3 \sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 4.4

Combine and simplify the denominator.

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Step 4.4.1

Multiply $\frac{7}{3 \sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 \sqrt{7} \sqrt{7}}$

Step 4.4.2

Move $\sqrt{7}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 (\sqrt{7} \sqrt{7})}$

Step 4.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 (\sqrt{7} \sqrt{7})}$

Step 4.4.4

Raise $\sqrt{7}$ to the power of $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 (\sqrt{7} \sqrt{7})}$

Step 4.4.5

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 {\sqrt{7}}^{1 + 1}}$

Step 4.4.6

Add $1$ and $1$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 {\sqrt{7}}^{2}}$

Step 4.4.7

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 {(7^{\frac{1}{2}})}^{2}}$

Step 4.4.7.2

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 \cdot 7^{\frac{1}{2} \cdot 2}}$

Step 4.4.7.3

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

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 \cdot 7^{\frac{2}{2}}}$

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.

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 \cdot 7^{\frac{2}{2}}}$

Step 4.4.7.4.2

Rewrite the expression.

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 \cdot 7}$

Step 4.4.7.5

Evaluate the exponent.

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 \cdot 7}$

Step 4.5

Cancel the common factor of $7$ .

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Step 4.5.1

Cancel the common factor.

$\cot (x) = \frac{1}{\tan (x)} = \frac{7 \sqrt{7}}{3 \cdot 7}$

Step 4.5.2

Rewrite the expression.

$\cot (x) = \frac{1}{\tan (x)} = \frac{\sqrt{7}}{3}$

Step 5

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{1}{\frac{\sqrt{7}}{4}}$

Step 6

Simplify the result.

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Step 6.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (x) = \frac{1}{\cos (x)} = 1 (\frac{4}{\sqrt{7}})$

Step 6.2

Multiply $\frac{4}{\sqrt{7}}$ by $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{4}{\sqrt{7}}$

Step 6.3

Multiply $\frac{4}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{4}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{4}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.4.2

Raise $\sqrt{7}$ to the power of $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.4.3

Raise $\sqrt{7}$ to the power of $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{\sqrt{7} \sqrt{7}}$

Step 6.4.4

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{1 + 1}}$

Step 6.4.5

Add $1$ and $1$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{{\sqrt{7}}^{2}}$

Step 6.4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{{(7^{\frac{1}{2}})}^{2}}$

Step 6.4.6.2

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{7^{\frac{1}{2} \cdot 2}}$

Step 6.4.6.3

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

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

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.

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{7^{\frac{2}{2}}}$

Step 6.4.6.4.2

Rewrite the expression.

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{7}$

Step 6.4.6.5

Evaluate the exponent.

$\sec (x) = \frac{1}{\cos (x)} = \frac{4 \sqrt{7}}{7}$

Step 7

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

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

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

Multiply $\frac{4}{3}$ by $1$ .

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

Step 9

The trig functions found are as follows:

$\sin (x) = \frac{3}{4}$

$\cos (x) = \frac{\sqrt{7}}{4}$

$\tan (x) = \frac{3 \sqrt{7}}{7}$

$\cot (x) = \frac{\sqrt{7}}{3}$

$\sec (x) = \frac{4 \sqrt{7}}{7}$

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