Find the Other Trig Values in Quadrant IV sin(x)=-4/5

$\sin (x) = - \frac{4}{5}$

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.

$Adjacent = \sqrt{{(5)}^{2} - {(- 4)}^{2}}$

Simplify inside the radical.

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

Adjacent $= \sqrt{25 - {(- 4)}^{2}}$

Raise $- 4$ to the power of $2$ .

Adjacent $= \sqrt{25 - 1 \cdot 16}$

Multiply $- 1$ by $16$ .

Adjacent $= \sqrt{25 - 16}$

Subtract $16$ from $25$ .

Adjacent $= \sqrt{9}$

Rewrite $9$ as $3^{2}$ .

Adjacent $= \sqrt{3^{2}}$

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

Adjacent $= 3$

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.

$\cos (x) = \frac{3}{5}$

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.

$\tan (x) = \frac{- 4}{3}$

Move the negative in front of the fraction.

$\tan (x) = - \frac{4}{3}$

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.

$\cot (x) = \frac{3}{- 4}$

Move the negative in front of the fraction.

$\cot (x) = - \frac{3}{4}$

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.

$\sec (x) = \frac{5}{3}$

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.

$\csc (x) = \frac{5}{- 4}$

Move the negative in front of the fraction.

$\csc (x) = - \frac{5}{4}$

This is the solution to each trig value.

$\sin (x) = - \frac{4}{5}$

$\cos (x) = \frac{3}{5}$

$\tan (x) = - \frac{4}{3}$

$\cot (x) = - \frac{3}{4}$

$\sec (x) = \frac{5}{3}$

$\csc (x) = - \frac{5}{4}$

Find the Other Trig Values in Quadrant IV sin(x)=-4/5

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