Find the Other Trig Values in Quadrant II tan(t)=-5/12

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

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.

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

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.

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

Step 4

Simplify inside the radical.

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

Hypotenuse $= \sqrt{25 + {(- 12)}^{2}}$

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

Hypotenuse $= \sqrt{25 + 144}$

Add $25$ and $144$ .

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 $\sin (t)$ .

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

Substitute in the known values.

$\sin (t) = \frac{5}{13}$

Step 6

Find the value of cosine.

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

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

Substitute in the known values.

$\cos (t) = \frac{- 12}{13}$

Move the negative in front of the fraction.

$\cos (t) = - \frac{12}{13}$

Step 7

Find the value of cotangent.

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

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

Substitute in the known values.

$\cot (t) = \frac{- 12}{5}$

Move the negative in front of the fraction.

$\cot (t) = - \frac{12}{5}$

Step 8

Find the value of secant.

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

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

Substitute in the known values.

$\sec (t) = \frac{13}{- 12}$

Move the negative in front of the fraction.

$\sec (t) = - \frac{13}{12}$

Step 9

Find the value of cosecant.

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

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

Substitute in the known values.

$\csc (t) = \frac{13}{5}$

Step 10

This is the solution to each trig value.

$\sin (t) = \frac{5}{13}$

$\cos (t) = - \frac{12}{13}$

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

$\cot (t) = - \frac{12}{5}$

$\sec (t) = - \frac{13}{12}$

$\csc (t) = \frac{13}{5}$

Find the Other Trig Values in Quadrant II tan(t)=-5/12

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