Find the Trig Value cos(theta)=-3/5 , pi/2<theta<pi

$\cos (θ) = - \frac{3}{5}$ , $\frac{π}{2} < θ < π$

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

$\cos (θ) = \frac{adjacent}{hypotenuse}$

Find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Replace the known values in the equation.

$Opposite = \sqrt{{(5)}^{2} - {(- 3)}^{2}}$

Simplify inside the radical.

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

Opposite $= \sqrt{25 - {(- 3)}^{2}}$

Raise $- 3$ to the power of $2$ .

Opposite $= \sqrt{25 - 1 \cdot 9}$

Multiply $- 1$ by $9$ .

Opposite $= \sqrt{25 - 9}$

Subtract $9$ from $25$ .

Opposite $= \sqrt{16}$

Rewrite $16$ as $4^{2}$ .

Opposite $= \sqrt{4^{2}}$

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

Opposite $= 4$

Find the value of sine.

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

$\sin (θ) = \frac{opp}{hyp}$

Substitute in the known values.

$\sin (θ) = \frac{4}{5}$

Find the value of tangent.

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

$\tan (θ) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ) = \frac{4}{- 3}$

Move the negative in front of the fraction.

$\tan (θ) = - \frac{4}{3}$

Find the value of cotangent.

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

$\cot (θ) = \frac{adj}{opp}$

Substitute in the known values.

$\cot (θ) = \frac{- 3}{4}$

Move the negative in front of the fraction.

$\cot (θ) = - \frac{3}{4}$

Find the value of secant.

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

$\sec (θ) = \frac{hyp}{adj}$

Substitute in the known values.

$\sec (θ) = \frac{5}{- 3}$

Move the negative in front of the fraction.

$\sec (θ) = - \frac{5}{3}$

Find the value of cosecant.

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

$\csc (θ) = \frac{hyp}{opp}$

Substitute in the known values.

$\csc (θ) = \frac{5}{4}$

This is the solution to each trig value.

$\sin (θ) = \frac{4}{5}$

$\cos (θ) = - \frac{3}{5}$

$\tan (θ) = - \frac{4}{3}$

$\cot (θ) = - \frac{3}{4}$

$\sec (θ) = - \frac{5}{3}$

$\csc (θ) = \frac{5}{4}$

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