Find the Other Trig Values in Quadrant III sin(0)=-5/13

$\sin (0) = - \frac{5}{13}$

Step 1

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.

$\sin (0) = \frac{opposite}{hypotenuse}$

Step 2

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.

$Adjacent = - \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

$Adjacent = - \sqrt{{(13)}^{2} - {(- 5)}^{2}}$

Step 4

Simplify inside the radical.

Negate $\sqrt{{(13)}^{2} - {(- 5)}^{2}}$ .

Adjacent $= - \sqrt{{(13)}^{2} - {(- 5)}^{2}}$

Raise $13$ to the power of $2$ .

Adjacent $= - \sqrt{169 - {(- 5)}^{2}}$

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

Adjacent $= - \sqrt{169 - 1 \cdot 25}$

Multiply $- 1$ by $25$ .

Adjacent $= - \sqrt{169 - 25}$

Subtract $25$ from $169$ .

Adjacent $= - \sqrt{144}$

Rewrite $144$ as $12^{2}$ .

Adjacent $= - \sqrt{12^{2}}$

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

Adjacent $= - 1 \cdot 12$

Multiply $- 1$ by $12$ .

Adjacent $= - 12$

Step 5

Find the value of cosine.

Use the definition of cosine to find the value of $\cos (0)$ .

$\cos (0) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (0) = \frac{- 12}{13}$

Move the negative in front of the fraction.

$\cos (0) = - \frac{12}{13}$

Step 6

Find the value of tangent.

Use the definition of tangent to find the value of $\tan (0)$ .

$\tan (0) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (0) = \frac{- 5}{- 12}$

Dividing two negative values results in a positive value.

$\tan (0) = \frac{5}{12}$

Step 7

Find the value of cotangent.

Use the definition of cotangent to find the value of $\cot (0)$ .

$\cot (0) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (0) = \frac{- 12}{- 5}$

Dividing two negative values results in a positive value.

$\cot (0) = \frac{12}{5}$

Step 8

Find the value of secant.

Use the definition of secant to find the value of $\sec (0)$ .

$\sec (0) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (0) = \frac{13}{- 12}$

Move the negative in front of the fraction.

$\sec (0) = - \frac{13}{12}$

Step 9

Find the value of cosecant.

Use the definition of cosecant to find the value of $\csc (0)$ .

$\csc (0) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (0) = \frac{13}{- 5}$

Move the negative in front of the fraction.

$\csc (0) = - \frac{13}{5}$

Step 10

This is the solution to each trig value.

$\sin (0) = - \frac{5}{13}$

$\cos (0) = - \frac{12}{13}$

$\tan (0) = \frac{5}{12}$

$\cot (0) = \frac{12}{5}$

$\sec (0) = - \frac{13}{12}$

$\csc (0) = - \frac{13}{5}$

Find the Other Trig Values in Quadrant III sin(0)=-5/13

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