Find the Other Trig Values in Quadrant II sin(x)=8/17

$\sin (x) = \frac{8}{17}$

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.

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

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.

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

Replace the known values in the equation.

$Adjacent = - \sqrt{{(17)}^{2} - {(8)}^{2}}$

Simplify inside the radical.

Negate $\sqrt{{(17)}^{2} - {(8)}^{2}}$ .

Adjacent $= - \sqrt{{(17)}^{2} - {(8)}^{2}}$

Raise $17$ to the power of $2$ .

Adjacent $= - \sqrt{289 - {(8)}^{2}}$

Raise $8$ to the power of $2$ .

Adjacent $= - \sqrt{289 - 1 \cdot 64}$

Multiply $- 1$ by $64$ .

Adjacent $= - \sqrt{289 - 64}$

Subtract $64$ from $289$ .

Adjacent $= - \sqrt{225}$

Rewrite $225$ as $15^{2}$ .

Adjacent $= - \sqrt{15^{2}}$

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

Adjacent $= - 1 \cdot 15$

Multiply $- 1$ by $15$ .

Adjacent $= - 15$

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.

$\cos (x) = \frac{- 15}{17}$

Move the negative in front of the fraction.

$\cos (x) = - \frac{15}{17}$

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.

$\tan (x) = \frac{8}{- 15}$

Move the negative in front of the fraction.

$\tan (x) = - \frac{8}{15}$

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.

$\cot (x) = \frac{- 15}{8}$

Move the negative in front of the fraction.

$\cot (x) = - \frac{15}{8}$

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.

$\sec (x) = \frac{17}{- 15}$

Move the negative in front of the fraction.

$\sec (x) = - \frac{17}{15}$

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.

$\csc (x) = \frac{17}{8}$

This is the solution to each trig value.

$\sin (x) = \frac{8}{17}$

$\cos (x) = - \frac{15}{17}$

$\tan (x) = - \frac{8}{15}$

$\cot (x) = - \frac{15}{8}$

$\sec (x) = - \frac{17}{15}$

$\csc (x) = \frac{17}{8}$

Find the Other Trig Values in Quadrant II sin(x)=8/17

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