Find the Other Trig Values in Quadrant IV sec(theta)=(2 square root of 3)/3

$\sec (θ) = \frac{2 \sqrt{3}}{3}$

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.

$Opposite = - \sqrt{{(2 \sqrt{3})}^{2} - {(3)}^{2}}$

Simplify inside the radical.

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Negate $\sqrt{{(2 \sqrt{3})}^{2} - {(3)}^{2}}$ .

Opposite $= - \sqrt{{(2 \sqrt{3})}^{2} - {(3)}^{2}}$

Apply the product rule to $2 \sqrt{3}$ .

Opposite $= - \sqrt{2^{2} {\sqrt{3}}^{2} - {(3)}^{2}}$

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

Opposite $= - \sqrt{4 {\sqrt{3}}^{2} - {(3)}^{2}}$

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

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

Opposite $= - \sqrt{4 {(3^{\frac{1}{2}})}^{2} - {(3)}^{2}}$

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

Opposite $= - \sqrt{4 \cdot 3^{\frac{1}{2} \cdot 2} - {(3)}^{2}}$

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

Opposite $= - \sqrt{4 \cdot 3^{\frac{2}{2}} - {(3)}^{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

Opposite $= - \sqrt{4 \cdot 3^{\frac{2}{2}} - {(3)}^{2}}$

Rewrite the expression.

Opposite $= - \sqrt{4 \cdot 3 - {(3)}^{2}}$

Evaluate the exponent.

Opposite $= - \sqrt{4 \cdot 3 - {(3)}^{2}}$

Multiply $4$ by $3$ .

Opposite $= - \sqrt{12 - {(3)}^{2}}$

Raise $3$ to the power of $2$ .

Opposite $= - \sqrt{12 - 1 \cdot 9}$

Multiply $- 1$ by $9$ .

Opposite $= - \sqrt{12 - 9}$

Subtract $9$ from $12$ .

Opposite $= - \sqrt{3}$

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.

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

Simplify the value of $\sin (θ)$ .

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Cancel the common factor of $\sqrt{3}$ .

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Cancel the common factor.

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

Rewrite the expression.

$\sin (θ) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$\sin (θ) = - \frac{1}{2}$

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.

$\cos (θ) = \frac{3}{2 \sqrt{3}}$

Simplify the value of $\cos (θ)$ .

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

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

Combine and simplify the denominator.

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

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

Move $\sqrt{3}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

$\cos (θ) = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cos (θ) = \frac{3 \sqrt{3}}{2 \cdot 3^{\frac{2}{2}}}$

Rewrite the expression.

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

Evaluate the exponent.

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

Cancel the common factor of $3$ .

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Cancel the common factor.

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

Rewrite the expression.

$\cos (θ) = \frac{\sqrt{3}}{2}$

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.

$\tan (θ) = \frac{- \sqrt{3}}{3}$

Move the negative in front of the fraction.

$\tan (θ) = - \frac{\sqrt{3}}{3}$

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.

$\cot (θ) = \frac{3}{- \sqrt{3}}$

Simplify the value of $\cot (θ)$ .

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Move the negative in front of the fraction.

$\cot (θ) = - \frac{3}{\sqrt{3}}$

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

$\cot (θ) = - \frac{3 \sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

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

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

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

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

$\cot (θ) = - \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\cot (θ) = - \frac{3 \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (θ) = - \frac{3 \sqrt{3}}{3}$

Evaluate the exponent.

$\cot (θ) = - \frac{3 \sqrt{3}}{3}$

Cancel the common factor of $3$ .

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Cancel the common factor.

$\cot (θ) = - \frac{3 \sqrt{3}}{3}$

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

$\cot (θ) = - \sqrt{3}$

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.

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

Simplify the value of $\csc (θ)$ .

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Cancel the common factor of $\sqrt{3}$ .

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Cancel the common factor.

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

Rewrite the expression.

$\csc (θ) = \frac{2}{- 1}$

Move the negative one from the denominator of $\frac{2}{- 1}$ .

$\csc (θ) = - 1 \cdot 2$

Multiply $- 1$ by $2$ .

$\csc (θ) = - 2$

This is the solution to each trig value.

$\sin (θ) = - \frac{1}{2}$

$\cos (θ) = \frac{\sqrt{3}}{2}$

$\tan (θ) = - \frac{\sqrt{3}}{3}$

$\cot (θ) = - \sqrt{3}$

$\sec (θ) = \frac{2 \sqrt{3}}{3}$

$\csc (θ) = - 2$

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