Find the Other Trig Values in Quadrant III sin(x)=-11/61

$\sin (x) = - \frac{11}{61}$

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.

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

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.

$Adjacent = - \sqrt{{(61)}^{2} - {(- 11)}^{2}}$

Simplify $- \sqrt{{(61)}^{2} - {(- 11)}^{2}}$ .

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Raise $61$ to the power of $2$ .

Adjacent $= - \sqrt{3721 - {(- 11)}^{2}}$

Raise $- 11$ to the power of $2$ .

Adjacent $= - \sqrt{3721 - 1 \cdot 121}$

Multiply $- 1$ by $121$ .

Adjacent $= - \sqrt{3721 - 121}$

Subtract $121$ from $3721$ .

Adjacent $= - \sqrt{3600}$

Rewrite $3600$ as $60^{2}$ .

Adjacent $= - \sqrt{60^{2}}$

Multiply.

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

Adjacent $= - 1 \cdot 60$

Multiply $- 1$ by $60$ .

Adjacent $= - 60$

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.

$\cos (x) = \frac{- 60}{61}$

Move the negative in front of the fraction.

$\cos (x) = - \frac{60}{61}$

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.

$\tan (x) = \frac{- 11}{- 60}$

Dividing two negative values results in a positive value.

$\tan (x) = \frac{11}{60}$

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.

$\cot (x) = \frac{- 60}{- 11}$

Dividing two negative values results in a positive value.

$\cot (x) = \frac{60}{11}$

Find the value of secant.

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

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

Substitute in the known values.

$\sec (x) = \frac{61}{- 60}$

Move the negative in front of the fraction.

$\sec (x) = - \frac{61}{60}$

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.

$\csc (x) = \frac{61}{- 11}$

Move the negative in front of the fraction.

$\csc (x) = - \frac{61}{11}$

This is the solution to each trig value.

$\sin (x) = - \frac{11}{61}$

$\cos (x) = - \frac{60}{61}$

$\tan (x) = \frac{11}{60}$

$\cot (x) = \frac{60}{11}$

$\sec (x) = - \frac{61}{60}$

$\csc (x) = - \frac{61}{11}$

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