Find the Other Trig Values in Quadrant III cot(x)=5/12

$\cot (x) = \frac{5}{12}$

Step 1

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

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

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.

$Hypotenuse = \sqrt{{(- 12)}^{2} + {(- 5)}^{2}}$

Step 4

Simplify inside the radical.

Raise $- 12$ to the power of $2$ .

Hypotenuse $= \sqrt{144 + {(- 5)}^{2}}$

Raise $- 5$ to the power of $2$ .

Hypotenuse $= \sqrt{144 + 25}$

Add $144$ and $25$ .

Hypotenuse $= \sqrt{169}$

Rewrite $169$ as $13^{2}$ .

Hypotenuse $= \sqrt{13^{2}}$

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

Hypotenuse $= 13$

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.

$\sin (x) = \frac{- 12}{13}$

Move the negative in front of the fraction.

$\sin (x) = - \frac{12}{13}$

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.

$\cos (x) = \frac{- 5}{13}$

Move the negative in front of the fraction.

$\cos (x) = - \frac{5}{13}$

Step 7

Find the value of tangent.

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

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

Substitute in the known values.

$\tan (x) = \frac{- 12}{- 5}$

Dividing two negative values results in a positive value.

$\tan (x) = \frac{12}{5}$

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.

$\sec (x) = \frac{13}{- 5}$

Move the negative in front of the fraction.

$\sec (x) = - \frac{13}{5}$

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.

$\csc (x) = \frac{13}{- 12}$

Move the negative in front of the fraction.

$\csc (x) = - \frac{13}{12}$

Step 10

This is the solution to each trig value.

$\sin (x) = - \frac{12}{13}$

$\cos (x) = - \frac{5}{13}$

$\tan (x) = \frac{12}{5}$

$\cot (x) = \frac{5}{12}$

$\sec (x) = - \frac{13}{5}$

$\csc (x) = - \frac{13}{12}$

Find the Other Trig Values in Quadrant III cot(x)=5/12

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