Find the Other Trig Values in Quadrant III sec(theta)=- square root of 5

$\sec (θ) = - \sqrt{5}$

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.

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

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.

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

Simplify inside the radical.

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Negate $\sqrt{{(\sqrt{5})}^{2} - {(- 1)}^{2}}$ .

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Multiply $- 1$ by ${(- 1)}^{2}$ by adding the exponents.

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Multiply $- 1$ by ${(- 1)}^{2}$ .

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Raise $- 1$ to the power of $1$ .

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

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

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

Add $1$ and $2$ .

Opposite $= - \sqrt{5 + {(- 1)}^{3}}$

Raise $- 1$ to the power of $3$ .

Opposite $= - \sqrt{5 - 1}$

Subtract $1$ from $5$ .

Opposite $= - \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

Opposite $= - 1 \cdot 2$

Multiply $- 1$ by $2$ .

Opposite $= - 2$

Find the value of sine.

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

$\sin (θ) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (θ) = \frac{- 2}{\sqrt{5}}$

Simplify the value of $\sin (θ)$ .

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Move the negative in front of the fraction.

$\sin (θ) = - \frac{2}{\sqrt{5}}$

Multiply $\frac{2}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = - (\frac{2}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

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Multiply $\frac{2}{\sqrt{5}}$ and $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sin (θ) = - \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

$\sin (θ) = - \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

$\sin (θ) = - \frac{2 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

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

$\sin (θ) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Add $1$ and $1$ .

$\sin (θ) = - \frac{2 \sqrt{5}}{{\sqrt{5}}^{2}}$

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

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

$\sin (θ) = - \frac{2 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

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

$\sin (θ) = - \frac{2 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

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

$\sin (θ) = - \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\sin (θ) = - \frac{2 \sqrt{5}}{5^{\frac{2}{2}}}$

Rewrite the expression.

$\sin (θ) = - \frac{2 \sqrt{5}}{5}$

Evaluate the exponent.

$\sin (θ) = - \frac{2 \sqrt{5}}{5}$

Find the value of cosine.

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

$\cos (θ) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (θ) = \frac{- 1}{\sqrt{5}}$

Simplify the value of $\cos (θ)$ .

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Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{\sqrt{5}}$

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\cos (θ) = - (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Combine and simplify the denominator.

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Multiply $\frac{1}{\sqrt{5}}$ and $\frac{\sqrt{5}}{\sqrt{5}}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{5}}{5}$

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{5}}{5}$

Find the value of tangent.

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

$\tan (θ) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ) = \frac{- 2}{- 1}$

Divide $- 2$ by $- 1$ .

$\tan (θ) = 2$

Find the value of cotangent.

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

$\cot (θ) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ) = \frac{- 1}{- 2}$

Dividing two negative values results in a positive value.

$\cot (θ) = \frac{1}{2}$

Find the value of cosecant.

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

$\csc (θ) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (θ) = \frac{\sqrt{5}}{- 2}$

Move the negative in front of the fraction.

$\csc (θ) = - \frac{\sqrt{5}}{2}$

This is the solution to each trig value.

$\sin (θ) = - \frac{2 \sqrt{5}}{5}$

$\cos (θ) = - \frac{\sqrt{5}}{5}$

$\tan (θ) = 2$

$\cot (θ) = \frac{1}{2}$

$\sec (θ) = - \sqrt{5}$

$\csc (θ) = - \frac{\sqrt{5}}{2}$

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