Find the Other Trig Values in Quadrant III tan(x)=20/21

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

Step 1

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.

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

Step 2

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.

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

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(- 20)}^{2} + {(- 21)}^{2}}$

Step 4

Simplify inside the radical.

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

Hypotenuse $= \sqrt{400 + {(- 21)}^{2}}$

Raise $- 21$ to the power of $2$ .

Hypotenuse $= \sqrt{400 + 441}$

Add $400$ and $441$ .

Hypotenuse $= \sqrt{841}$

Rewrite $841$ as $29^{2}$ .

Hypotenuse $= \sqrt{29^{2}}$

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

Hypotenuse $= 29$

Step 5

Find the value of sine.

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

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

Substitute in the known values.

$\sin (x) = \frac{- 20}{29}$

Move the negative in front of the fraction.

$\sin (x) = - \frac{20}{29}$

Step 6

Find the value of cosine.

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

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

Substitute in the known values.

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

Move the negative in front of the fraction.

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

Step 7

Find the value of cotangent.

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

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

Substitute in the known values.

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

Dividing two negative values results in a positive value.

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

Step 8

Find the value of secant.

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

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

Substitute in the known values.

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

Move the negative in front of the fraction.

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

Step 9

Find the value of cosecant.

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

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

Substitute in the known values.

$\csc (x) = \frac{29}{- 20}$

Move the negative in front of the fraction.

$\csc (x) = - \frac{29}{20}$

Step 10

This is the solution to each trig value.

$\sin (x) = - \frac{20}{29}$

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

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

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

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

$\csc (x) = - \frac{29}{20}$

Find the Other Trig Values in Quadrant III tan(x)=20/21

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