Trigonometry Examples

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

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.

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

Step 2

Simplify the result.

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Step 2.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $- 1 \cdot 2$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 2.3

Simplify the expression.

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Step 2.3.1

Move $- 1$ to the left of $\sqrt{3}$ .

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

Step 2.3.2

Rewrite $- 1 \sqrt{3}$ as $- \sqrt{3}$ .

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

Step 3

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

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

Step 4

Simplify the result.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

Move the negative in front of the fraction.

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

Step 4.2

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

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

Step 4.3

Combine and simplify the denominator.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

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

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

Step 4.3.5

Add $1$ and $1$ .

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

Step 4.3.6

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

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Step 4.3.6.1

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

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

Step 4.3.6.2

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

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

Step 4.3.6.3

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

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

Step 4.3.6.4

Cancel the common factor of $2$ .

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Step 4.3.6.4.1

Cancel the common factor.

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

Step 4.3.6.4.2

Rewrite the expression.

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

Step 4.3.6.5

Evaluate the exponent.

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

Step 5

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

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

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 $- 1 (- 1)$ .

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

Step 6.1.2

Move the negative in front of the fraction.

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

Step 6.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 6.3

Multiply $- (1 \cdot 2)$ .

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Step 6.3.1

Multiply $2$ by $1$ .

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

Step 6.3.2

Multiply $- 1$ by $2$ .

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

Step 7

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

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

Step 8

Simplify the result.

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Step 8.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Combine and simplify the denominator.

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Step 8.4.1

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

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

Step 8.4.2

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

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

Step 8.4.3

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

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

Step 8.4.4

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

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

Step 8.4.5

Add $1$ and $1$ .

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

Step 8.4.6

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

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Step 8.4.6.1

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

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

Step 8.4.6.2

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

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

Step 8.4.6.3

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

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

Step 8.4.6.4

Cancel the common factor of $2$ .

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Step 8.4.6.4.1

Cancel the common factor.

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

Step 8.4.6.4.2

Rewrite the expression.

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

Step 8.4.6.5

Evaluate the exponent.

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

Step 9

The trig functions found are as follows:

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

$\cos (x) = - \frac{1}{2}$

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

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

$\sec (x) = - 2$

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