Find the Other Trig Values in Quadrant III sin(theta)=-1/10

$\sin (θ) = - \frac{1}{10}$

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

$\sin (θ) = \frac{opposite}{hypotenuse}$

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

$Adjacent = - \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Replace the known values in the equation.

$Adjacent = - \sqrt{{(10)}^{2} - {(- 1)}^{2}}$

Simplify inside the radical.

Negate $\sqrt{{(10)}^{2} - {(- 1)}^{2}}$ .

Adjacent $= - \sqrt{{(10)}^{2} - {(- 1)}^{2}}$

Raise $10$ to the power of $2$ .

Adjacent $= - \sqrt{100 - {(- 1)}^{2}}$

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

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

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

Adjacent $= - \sqrt{100 + (- 1) {(- 1)}^{2}}$

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

Adjacent $= - \sqrt{100 + {(- 1)}^{1 + 2}}$

Add $1$ and $2$ .

Adjacent $= - \sqrt{100 + {(- 1)}^{3}}$

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

Adjacent $= - \sqrt{100 - 1}$

Subtract $1$ from $100$ .

Adjacent $= - \sqrt{99}$

Rewrite $99$ as $3^{2} \cdot 11$ .

Factor $9$ out of $99$ .

Adjacent $= - \sqrt{9 (11)}$

Rewrite $9$ as $3^{2}$ .

Adjacent $= - \sqrt{3^{2} \cdot 11}$

Pull terms out from under the radical.

Adjacent $= - (3 \sqrt{11})$

Multiply $3$ by $- 1$ .

Adjacent $= - 3 \sqrt{11}$

Find the value of cosine.

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

$\cos (θ) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (θ) = \frac{- 3 \sqrt{11}}{10}$

Move the negative in front of the fraction.

$\cos (θ) = - \frac{3 \sqrt{11}}{10}$

Find the value of tangent.

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

$\tan (θ) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ) = \frac{- 1}{- 3 \sqrt{11}}$

Simplify the value of $\tan (θ)$ .

Dividing two negative values results in a positive value.

$\tan (θ) = \frac{1}{(3) \sqrt{11}}$

Multiply $\frac{1}{3 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\tan (θ) = \frac{1}{3 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}}$

Combine and simplify the denominator.

Multiply $\frac{1}{3 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\tan (θ) = \frac{\sqrt{11}}{3 \sqrt{11} \sqrt{11}}$

Move $\sqrt{11}$ .

$\tan (θ) = \frac{\sqrt{11}}{3 (\sqrt{11} \sqrt{11})}$

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

$\tan (θ) = \frac{\sqrt{11}}{3 (\sqrt{11} \sqrt{11})}$

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

$\tan (θ) = \frac{\sqrt{11}}{3 (\sqrt{11} \sqrt{11})}$

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

$\tan (θ) = \frac{\sqrt{11}}{3 {\sqrt{11}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (θ) = \frac{\sqrt{11}}{3 {\sqrt{11}}^{2}}$

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

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

$\tan (θ) = \frac{\sqrt{11}}{3 {(11^{\frac{1}{2}})}^{2}}$

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

$\tan (θ) = \frac{\sqrt{11}}{3 \cdot 11^{\frac{1}{2} \cdot 2}}$

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

$\tan (θ) = \frac{\sqrt{11}}{3 \cdot 11^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (θ) = \frac{\sqrt{11}}{3 \cdot 11^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (θ) = \frac{\sqrt{11}}{3 \cdot 11}$

Evaluate the exponent.

$\tan (θ) = \frac{\sqrt{11}}{3 \cdot 11}$

Multiply $3$ by $11$ .

$\tan (θ) = \frac{\sqrt{11}}{33}$

Find the value of cotangent.

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

$\cot (θ) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ) = \frac{- 3 \sqrt{11}}{- 1}$

Simplify the value of $\cot (θ)$ .

Move the negative one from the denominator of $\frac{- 3 \sqrt{11}}{- 1}$ .

$\cot (θ) = - 1 \cdot (- 3 \sqrt{11})$

Rewrite $- 1 \cdot (- 3 \sqrt{11})$ as $- (- 3 \sqrt{11})$ .

$\cot (θ) = - (- 3 \sqrt{11})$

Multiply $- 3$ by $- 1$ .

$\cot (θ) = 3 \sqrt{11}$

Find the value of secant.

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

$\sec (θ) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (θ) = \frac{10}{- 3 \sqrt{11}}$

Simplify the value of $\sec (θ)$ .

Move the negative in front of the fraction.

$\sec (θ) = - \frac{10}{(3) \sqrt{11}}$

Multiply $\frac{10}{3 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\sec (θ) = - (\frac{10}{3 \sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}})$

Combine and simplify the denominator.

Multiply $\frac{10}{3 \sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\sec (θ) = - \frac{10 \sqrt{11}}{3 \sqrt{11} \sqrt{11}}$

Move $\sqrt{11}$ .

$\sec (θ) = - \frac{10 \sqrt{11}}{3 (\sqrt{11} \sqrt{11})}$

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

$\sec (θ) = - \frac{10 \sqrt{11}}{3 (\sqrt{11} \sqrt{11})}$

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

$\sec (θ) = - \frac{10 \sqrt{11}}{3 (\sqrt{11} \sqrt{11})}$

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

$\sec (θ) = - \frac{10 \sqrt{11}}{3 {\sqrt{11}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (θ) = - \frac{10 \sqrt{11}}{3 {\sqrt{11}}^{2}}$

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

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

$\sec (θ) = - \frac{10 \sqrt{11}}{3 {(11^{\frac{1}{2}})}^{2}}$

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

$\sec (θ) = - \frac{10 \sqrt{11}}{3 \cdot 11^{\frac{1}{2} \cdot 2}}$

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

$\sec (θ) = - \frac{10 \sqrt{11}}{3 \cdot 11^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (θ) = - \frac{10 \sqrt{11}}{3 \cdot 11^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (θ) = - \frac{10 \sqrt{11}}{3 \cdot 11}$

Evaluate the exponent.

$\sec (θ) = - \frac{10 \sqrt{11}}{3 \cdot 11}$

Multiply $3$ by $11$ .

$\sec (θ) = - \frac{10 \sqrt{11}}{33}$

Find the value of cosecant.

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

$\csc (θ) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (θ) = \frac{10}{- 1}$

Divide $10$ by $- 1$ .

$\csc (θ) = - 10$

This is the solution to each trig value.

$\sin (θ) = - \frac{1}{10}$

$\cos (θ) = - \frac{3 \sqrt{11}}{10}$

$\tan (θ) = \frac{\sqrt{11}}{33}$

$\cot (θ) = 3 \sqrt{11}$

$\sec (θ) = - \frac{10 \sqrt{11}}{33}$

$\csc (θ) = - 10$

Find the Other Trig Values in Quadrant III sin(theta)=-1/10

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