Find the Other Trig Values in Quadrant IV cos(x)=20/29

$\cos (x) = \frac{20}{29}$

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

$\cos (x) = \frac{adjacent}{hypotenuse}$

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.

$Opposite = - \sqrt{{(29)}^{2} - {(20)}^{2}}$

Simplify $- \sqrt{{(29)}^{2} - {(20)}^{2}}$ .

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Raise $29$ to the power of $2$ .

Opposite $= - \sqrt{841 - {(20)}^{2}}$

Raise $20$ to the power of $2$ .

Opposite $= - \sqrt{841 - 1 \cdot 400}$

Multiply $- 1$ by $400$ .

Opposite $= - \sqrt{841 - 400}$

Subtract $400$ from $841$ .

Opposite $= - \sqrt{441}$

Rewrite $441$ as $21^{2}$ .

Opposite $= - \sqrt{21^{2}}$

Multiply.

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

Opposite $= - 1 \cdot 21$

Multiply $- 1$ by $21$ .

Opposite $= - 21$

Find the value of sine.

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

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

Substitute in the known values.

$\sin (x) = \frac{- 21}{29}$

Move the negative in front of the fraction.

$\sin (x) = - \frac{21}{29}$

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.

$\tan (x) = \frac{- 21}{20}$

Move the negative in front of the fraction.

$\tan (x) = - \frac{21}{20}$

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.

$\cot (x) = \frac{20}{- 21}$

Move the negative in front of the fraction.

$\cot (x) = - \frac{20}{21}$

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.

$\sec (x) = \frac{29}{20}$

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.

$\csc (x) = \frac{29}{- 21}$

Move the negative in front of the fraction.

$\csc (x) = - \frac{29}{21}$

This is the solution to each trig value.

$\sin (x) = - \frac{21}{29}$

$\cos (x) = \frac{20}{29}$

$\tan (x) = - \frac{21}{20}$

$\cot (x) = - \frac{20}{21}$

$\sec (x) = \frac{29}{20}$

$\csc (x) = - \frac{29}{21}$

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