Find the Other Trig Values in Quadrant III tan(0) = square root of 3

$\tan (0) = \sqrt{3}$

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

$\tan (0) = \frac{opposite}{adjacent}$

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

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(- \sqrt{3})}^{2} + {(- 1)}^{2}}$

Simplify inside the radical.

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

Hypotenuse $= \sqrt{{(- 1)}^{2} {\sqrt{3}}^{2} + {(- 1)}^{2}}$

Raise $- 1$ to the power of $2$ .

Hypotenuse $= \sqrt{1 {\sqrt{3}}^{2} + {(- 1)}^{2}}$

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

Hypotenuse $= \sqrt{{\sqrt{3}}^{2} + {(- 1)}^{2}}$

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

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

Hypotenuse $= \sqrt{{(3^{\frac{1}{2}})}^{2} + {(- 1)}^{2}}$

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

Hypotenuse $= \sqrt{3^{\frac{1}{2} \cdot 2} + {(- 1)}^{2}}$

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

Hypotenuse $= \sqrt{3^{\frac{2}{2}} + {(- 1)}^{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

Hypotenuse $= \sqrt{3^{\frac{2}{2}} + {(- 1)}^{2}}$

Rewrite the expression.

Hypotenuse $= \sqrt{3 + {(- 1)}^{2}}$

Evaluate the exponent.

Hypotenuse $= \sqrt{3 + {(- 1)}^{2}}$

Raise $- 1$ to the power of $2$ .

Hypotenuse $= \sqrt{3 + 1}$

Add $3$ and $1$ .

Hypotenuse $= \sqrt{4}$

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

Hypotenuse $= \sqrt{2^{2}}$

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

Hypotenuse $= 2$

Find the value of sine.

Use the definition of sine to find the value of $\sin (0)$ .

$\sin (0) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (0) = \frac{- \sqrt{3}}{2}$

Move the negative in front of the fraction.

$\sin (0) = - \frac{\sqrt{3}}{2}$

Find the value of cosine.

Use the definition of cosine to find the value of $\cos (0)$ .

$\cos (0) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (0) = \frac{- 1}{2}$

Move the negative in front of the fraction.

$\cos (0) = - \frac{1}{2}$

Find the value of cotangent.

Use the definition of cotangent to find the value of $\cot (0)$ .

$\cot (0) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (0) = \frac{- 1}{- \sqrt{3}}$

Simplify the value of $\cot (0)$ .

Dividing two negative values results in a positive value.

$\cot (0) = \frac{1}{\sqrt{3}}$

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

$\cot (0) = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Combine and simplify the denominator.

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

$\cot (0) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\cot (0) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\cot (0) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\cot (0) = \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\cot (0) = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$\cot (0) = \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\cot (0) = \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$\cot (0) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\cot (0) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$\cot (0) = \frac{\sqrt{3}}{3}$

Evaluate the exponent.

$\cot (0) = \frac{\sqrt{3}}{3}$

Find the value of secant.

Use the definition of secant to find the value of $\sec (0)$ .

$\sec (0) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (0) = \frac{2}{- 1}$

Divide $2$ by $- 1$ .

$\sec (0) = - 2$

Find the value of cosecant.

Use the definition of cosecant to find the value of $\csc (0)$ .

$\csc (0) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (0) = \frac{2}{- \sqrt{3}}$

Simplify the value of $\csc (0)$ .

Move the negative in front of the fraction.

$\csc (0) = - \frac{2}{\sqrt{3}}$

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

$\csc (0) = - (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Combine and simplify the denominator.

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

$\csc (0) = - \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\csc (0) = - \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\csc (0) = - \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

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

$\csc (0) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Add $1$ and $1$ .

$\csc (0) = - \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

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

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

$\csc (0) = - \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

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

$\csc (0) = - \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

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

$\csc (0) = - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\csc (0) = - \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Rewrite the expression.

$\csc (0) = - \frac{2 \sqrt{3}}{3}$

Evaluate the exponent.

$\csc (0) = - \frac{2 \sqrt{3}}{3}$

This is the solution to each trig value.

$\sin (0) = - \frac{\sqrt{3}}{2}$

$\cos (0) = - \frac{1}{2}$

$\tan (0) = \sqrt{3}$

$\cot (0) = \frac{\sqrt{3}}{3}$

$\sec (0) = - 2$

$\csc (0) = - \frac{2 \sqrt{3}}{3}$

Find the Other Trig Values in Quadrant III tan(0) = square root of 3

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