Precalculus Examples

$\sec (x) = - \frac{4 \sqrt{5}}{5}$

Use the definition of secant to find the known sides of the unit circle right triangle. The quadrant determines the sign on each of the values.

$\sec (x) = \frac{hypotenuse}{adjacent}$

Find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Replace the known values in the equation.

$Opposite = \sqrt{{(4 \sqrt{5})}^{2} - {(- 5)}^{2}}$

Simplify inside the radical.

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Apply the product rule to $4 \sqrt{5}$ .

Opposite $= \sqrt{4^{2} {\sqrt{5}}^{2} - {(- 5)}^{2}}$

Raise $4$ to the power of $2$ .

Opposite $= \sqrt{16 {\sqrt{5}}^{2} - {(- 5)}^{2}}$

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

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

Opposite $= \sqrt{16 {(5^{\frac{1}{2}})}^{2} - {(- 5)}^{2}}$

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

Opposite $= \sqrt{16 \cdot 5^{\frac{1}{2} \cdot 2} - {(- 5)}^{2}}$

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

Opposite $= \sqrt{16 \cdot 5^{\frac{2}{2}} - {(- 5)}^{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

Opposite $= \sqrt{16 \cdot 5^{\frac{2}{2}} - {(- 5)}^{2}}$

Rewrite the expression.

Opposite $= \sqrt{16 \cdot 5 - {(- 5)}^{2}}$

Evaluate the exponent.

Opposite $= \sqrt{16 \cdot 5 - {(- 5)}^{2}}$

Multiply $16$ by $5$ .

Opposite $= \sqrt{80 - {(- 5)}^{2}}$

Raise $- 5$ to the power of $2$ .

Opposite $= \sqrt{80 - 1 \cdot 25}$

Multiply $- 1$ by $25$ .

Opposite $= \sqrt{80 - 25}$

Subtract $25$ from $80$ .

Opposite $= \sqrt{55}$

Find the value of sine.

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Use the definition of sine to find the value of $\sin (x)$ .

$\sin (x) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (x) = \frac{\sqrt{55}}{4 \sqrt{5}}$

Simplify the value of $\sin (x)$ .

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Combine $\sqrt{55}$ and $\sqrt{5}$ into a single radical.

$\sin (x) = \frac{\sqrt{\frac{55}{5}}}{4}$

Cancel the common factor of $55$ and $5$ .

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Factor $5$ out of $55$ .

$\sin (x) = \frac{\sqrt{\frac{5 \cdot 11}{5}}}{4}$

Cancel the common factors.

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Factor $5$ out of $5$ .

$\sin (x) = \frac{\sqrt{\frac{5 \cdot 11}{5 (1)}}}{4}$

Cancel the common factor.

$\sin (x) = \frac{\sqrt{\frac{5 \cdot 11}{5 \cdot 1}}}{4}$

Rewrite the expression.

$\sin (x) = \frac{\sqrt{\frac{11}{1}}}{4}$

Divide $11$ by $1$ .

$\sin (x) = \frac{\sqrt{11}}{4}$

Find the value of cosine.

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Use the definition of cosine to find the value of $\cos (x)$ .

$\cos (x) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (x) = \frac{- 5}{4 \sqrt{5}}$

Simplify the value of $\cos (x)$ .

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Move the negative in front of the fraction.

$\cos (x) = - \frac{5}{4 \sqrt{5}}$

Multiply $\frac{5}{4 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\cos (x) = - (\frac{5}{4 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

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Multiply $\frac{5}{4 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\cos (x) = - \frac{5 \sqrt{5}}{4 \sqrt{5} \sqrt{5}}$

Move $\sqrt{5}$ .

$\cos (x) = - \frac{5 \sqrt{5}}{4 (\sqrt{5} \sqrt{5})}$

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

$\cos (x) = - \frac{5 \sqrt{5}}{4 (\sqrt{5} \sqrt{5})}$

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

$\cos (x) = - \frac{5 \sqrt{5}}{4 (\sqrt{5} \sqrt{5})}$

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

$\cos (x) = - \frac{5 \sqrt{5}}{4 {\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$\cos (x) = - \frac{5 \sqrt{5}}{4 {\sqrt{5}}^{2}}$

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

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

$\cos (x) = - \frac{5 \sqrt{5}}{4 {(5^{\frac{1}{2}})}^{2}}$

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

$\cos (x) = - \frac{5 \sqrt{5}}{4 \cdot 5^{\frac{1}{2} \cdot 2}}$

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

$\cos (x) = - \frac{5 \sqrt{5}}{4 \cdot 5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cos (x) = - \frac{5 \sqrt{5}}{4 \cdot 5^{\frac{2}{2}}}$

Rewrite the expression.

$\cos (x) = - \frac{5 \sqrt{5}}{4 \cdot 5}$

Evaluate the exponent.

$\cos (x) = - \frac{5 \sqrt{5}}{4 \cdot 5}$

Cancel the common factor of $5$ .

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Cancel the common factor.

$\cos (x) = - \frac{5 \sqrt{5}}{4 \cdot 5}$

Rewrite the expression.

$\cos (x) = - \frac{\sqrt{5}}{4}$

Find the value of tangent.

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Use the definition of tangent to find the value of $\tan (x)$ .

$\tan (x) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (x) = \frac{\sqrt{55}}{- 5}$

Move the negative in front of the fraction.

$\tan (x) = - \frac{\sqrt{55}}{5}$

Find the value of cotangent.

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Use the definition of cotangent to find the value of $\cot (x)$ .

$\cot (x) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (x) = \frac{- 5}{\sqrt{55}}$

Simplify the value of $\cot (x)$ .

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Move the negative in front of the fraction.

$\cot (x) = - \frac{5}{\sqrt{55}}$

Multiply $\frac{5}{\sqrt{55}}$ by $\frac{\sqrt{55}}{\sqrt{55}}$ .

$\cot (x) = - (\frac{5}{\sqrt{55}} \cdot \frac{\sqrt{55}}{\sqrt{55}})$

Combine and simplify the denominator.

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Multiply $\frac{5}{\sqrt{55}}$ by $\frac{\sqrt{55}}{\sqrt{55}}$ .

$\cot (x) = - \frac{5 \sqrt{55}}{\sqrt{55} \sqrt{55}}$

Raise $\sqrt{55}$ to the power of $1$ .

$\cot (x) = - \frac{5 \sqrt{55}}{\sqrt{55} \sqrt{55}}$

Raise $\sqrt{55}$ to the power of $1$ .

$\cot (x) = - \frac{5 \sqrt{55}}{\sqrt{55} \sqrt{55}}$

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

$\cot (x) = - \frac{5 \sqrt{55}}{{\sqrt{55}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (x) = - \frac{5 \sqrt{55}}{{\sqrt{55}}^{2}}$

Rewrite ${\sqrt{55}}^{2}$ as $55$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{55}$ as $55^{\frac{1}{2}}$ .

$\cot (x) = - \frac{5 \sqrt{55}}{{(55^{\frac{1}{2}})}^{2}}$

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

$\cot (x) = - \frac{5 \sqrt{55}}{55^{\frac{1}{2} \cdot 2}}$

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

$\cot (x) = - \frac{5 \sqrt{55}}{55^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cot (x) = - \frac{5 \sqrt{55}}{55^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (x) = - \frac{5 \sqrt{55}}{55}$

Evaluate the exponent.

$\cot (x) = - \frac{5 \sqrt{55}}{55}$

Cancel the common factor of $5$ and $55$ .

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Factor $5$ out of $5 \sqrt{55}$ .

$\cot (x) = - \frac{5 (\sqrt{55})}{55}$

Cancel the common factors.

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Factor $5$ out of $55$ .

$\cot (x) = - \frac{5 \sqrt{55}}{5 \cdot 11}$

Cancel the common factor.

$\cot (x) = - \frac{5 \sqrt{55}}{5 \cdot 11}$

Rewrite the expression.

$\cot (x) = - \frac{\sqrt{55}}{11}$

Find the value of cosecant.

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Use the definition of cosecant to find the value of $\csc (x)$ .

$\csc (x) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (x) = \frac{4 \sqrt{5}}{\sqrt{55}}$

Simplify the value of $\csc (x)$ .

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Combine $\sqrt{5}$ and $\sqrt{55}$ into a single radical.

$\csc (x) = 4 \sqrt{\frac{5}{55}}$

Cancel the common factor of $5$ and $55$ .

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Factor $5$ out of $5$ .

$\csc (x) = 4 \sqrt{\frac{5 (1)}{55}}$

Cancel the common factors.

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Factor $5$ out of $55$ .

$\csc (x) = 4 \sqrt{\frac{5 \cdot 1}{5 \cdot 11}}$

Cancel the common factor.

$\csc (x) = 4 \sqrt{\frac{5 \cdot 1}{5 \cdot 11}}$

Rewrite the expression.

$\csc (x) = 4 \sqrt{\frac{1}{11}}$

Rewrite $\sqrt{\frac{1}{11}}$ as $\frac{\sqrt{1}}{\sqrt{11}}$ .

$\csc (x) = 4 (\frac{\sqrt{1}}{\sqrt{11}})$

Any root of $1$ is $1$ .

$\csc (x) = 4 (\frac{1}{\sqrt{11}})$

Multiply $\frac{1}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\csc (x) = 4 (\frac{1}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}})$

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\csc (x) = 4 (\frac{\sqrt{11}}{\sqrt{11} \sqrt{11}})$

Raise $\sqrt{11}$ to the power of $1$ .

$\csc (x) = 4 (\frac{\sqrt{11}}{\sqrt{11} \sqrt{11}})$

Raise $\sqrt{11}$ to the power of $1$ .

$\csc (x) = 4 (\frac{\sqrt{11}}{\sqrt{11} \sqrt{11}})$

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

$\csc (x) = 4 (\frac{\sqrt{11}}{{\sqrt{11}}^{1 + 1}})$

Add $1$ and $1$ .

$\csc (x) = 4 (\frac{\sqrt{11}}{{\sqrt{11}}^{2}})$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\csc (x) = 4 (\frac{\sqrt{11}}{{(11^{\frac{1}{2}})}^{2}})$

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

$\csc (x) = 4 (\frac{\sqrt{11}}{11^{\frac{1}{2} \cdot 2}})$

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

$\csc (x) = 4 (\frac{\sqrt{11}}{11^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\csc (x) = 4 (\frac{\sqrt{11}}{11^{\frac{2}{2}}})$

Rewrite the expression.

$\csc (x) = 4 (\frac{\sqrt{11}}{11})$

Evaluate the exponent.

$\csc (x) = 4 (\frac{\sqrt{11}}{11})$

Combine $4$ and $\frac{\sqrt{11}}{11}$ .

$\csc (x) = \frac{4 \sqrt{11}}{11}$

This is the solution to each trig value.

$\sin (x) = \frac{\sqrt{11}}{4}$

$\cos (x) = - \frac{\sqrt{5}}{4}$

$\tan (x) = - \frac{\sqrt{55}}{5}$

$\cot (x) = - \frac{\sqrt{55}}{11}$

$\sec (x) = - \frac{4 \sqrt{5}}{5}$

$\csc (x) = \frac{4 \sqrt{11}}{11}$