Find the Other Trig Values in Quadrant III sin(x)=-7/25

$\sin (x) = - \frac{7}{25}$

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.

$Adjacent = - \sqrt{{(25)}^{2} - {(- 7)}^{2}}$

Simplify inside the radical.

Negate $\sqrt{{(25)}^{2} - {(- 7)}^{2}}$ .

Adjacent $= - \sqrt{{(25)}^{2} - {(- 7)}^{2}}$

Raise $25$ to the power of $2$ .

Adjacent $= - \sqrt{625 - {(- 7)}^{2}}$

Raise $- 7$ to the power of $2$ .

Adjacent $= - \sqrt{625 - 1 \cdot 49}$

Multiply $- 1$ by $49$ .

Adjacent $= - \sqrt{625 - 49}$

Subtract $49$ from $625$ .

Adjacent $= - \sqrt{576}$

Rewrite $576$ as $24^{2}$ .

Adjacent $= - \sqrt{24^{2}}$

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

Adjacent $= - 1 \cdot 24$

Multiply $- 1$ by $24$ .

Adjacent $= - 24$

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.

$\cos (x) = \frac{- 24}{25}$

Move the negative in front of the fraction.

$\cos (x) = - \frac{24}{25}$

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.

$\tan (x) = \frac{- 7}{- 24}$

Dividing two negative values results in a positive value.

$\tan (x) = \frac{7}{24}$

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.

$\cot (x) = \frac{- 24}{- 7}$

Dividing two negative values results in a positive value.

$\cot (x) = \frac{24}{7}$

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.

$\sec (x) = \frac{25}{- 24}$

Move the negative in front of the fraction.

$\sec (x) = - \frac{25}{24}$

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.

$\csc (x) = \frac{25}{- 7}$

Move the negative in front of the fraction.

$\csc (x) = - \frac{25}{7}$

This is the solution to each trig value.

$\sin (x) = - \frac{7}{25}$

$\cos (x) = - \frac{24}{25}$

$\tan (x) = \frac{7}{24}$

$\cot (x) = \frac{24}{7}$

$\sec (x) = - \frac{25}{24}$

$\csc (x) = - \frac{25}{7}$

Find the Other Trig Values in Quadrant III sin(x)=-7/25

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