Find the Other Trig Values in Quadrant II csc(theta)=13/12

$\csc (θ) = \frac{13}{12}$

Step 1

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

$\csc (θ) = \frac{hypotenuse}{opposite}$

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.

$Adjacent = - \sqrt{{(13)}^{2} - {(12)}^{2}}$

Step 4

Simplify inside the radical.

Negate $\sqrt{{(13)}^{2} - {(12)}^{2}}$ .

Adjacent $= - \sqrt{{(13)}^{2} - {(12)}^{2}}$

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

Adjacent $= - \sqrt{169 - {(12)}^{2}}$

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

Adjacent $= - \sqrt{169 - 1 \cdot 144}$

Multiply $- 1$ by $144$ .

Adjacent $= - \sqrt{169 - 144}$

Subtract $144$ from $169$ .

Adjacent $= - \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

Adjacent $= - \sqrt{5^{2}}$

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

Adjacent $= - 1 \cdot 5$

Multiply $- 1$ by $5$ .

Adjacent $= - 5$

Step 5

Find the value of sine.

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

$\sin (θ) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (θ) = \frac{12}{13}$

Step 6

Find the value of cosine.

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

$\cos (θ) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (θ) = \frac{- 5}{13}$

Move the negative in front of the fraction.

$\cos (θ) = - \frac{5}{13}$

Step 7

Find the value of tangent.

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

$\tan (θ) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ) = \frac{12}{- 5}$

Move the negative in front of the fraction.

$\tan (θ) = - \frac{12}{5}$

Step 8

Find the value of cotangent.

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

$\cot (θ) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ) = \frac{- 5}{12}$

Move the negative in front of the fraction.

$\cot (θ) = - \frac{5}{12}$

Step 9

Find the value of secant.

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

$\sec (θ) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (θ) = \frac{13}{- 5}$

Move the negative in front of the fraction.

$\sec (θ) = - \frac{13}{5}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{12}{13}$

$\cos (θ) = - \frac{5}{13}$

$\tan (θ) = - \frac{12}{5}$

$\cot (θ) = - \frac{5}{12}$

$\sec (θ) = - \frac{13}{5}$

$\csc (θ) = \frac{13}{12}$

Find the Other Trig Values in Quadrant II csc(theta)=13/12

Related Questions

Support LCMGCF.com by disabling your adblocker.