Find the Other Trig Values in Quadrant III tan(theta)=0

$\tan (θ) = 0$

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

$\tan (θ) = \frac{opposite}{adjacent}$

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

$Hypotenuse = \sqrt{{opposite}^{2} + {adjacent}^{2}}$

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(- 0)}^{2} + {(- 1)}^{2}}$

Simplify inside the radical.

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Multiply $- 1$ by $0$ .

Hypotenuse $= \sqrt{0^{2} + {(- 1)}^{2}}$

Raising $0$ to any positive power yields $0$ .

Hypotenuse $= \sqrt{0 + {(- 1)}^{2}}$

Raise $- 1$ to the power of $2$ .

Hypotenuse $= \sqrt{0 + 1}$

Add $0$ and $1$ .

Hypotenuse $= \sqrt{1}$

Any root of $1$ is $1$ .

Hypotenuse $= 1$

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.

$\sin (θ) = \frac{- 0}{1}$

Simplify the value of $\sin (θ)$ .

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Divide $- 0$ by $1$ .

$\sin (θ) = - 0$

Multiply $- 1$ by $0$ .

$\sin (θ) = 0$

Find the value of cosine.

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

$\cos (θ) = \frac{adj}{hyp}$

Substitute in the known values.

$\cos (θ) = \frac{- 1}{1}$

Divide $- 1$ by $1$ .

$\cos (θ) = - 1$

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.

$\cot (θ) = \frac{- 1}{- 0}$

Division by $0$ results in cotangent being undefined at $θ$ .

$\cot (θ) = Undefined$

Undefined

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.

$\sec (θ) = \frac{1}{- 1}$

Divide $1$ by $- 1$ .

$\sec (θ) = - 1$

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.

$\csc (θ) = \frac{1}{- 0}$

Division by $0$ results in cosecant being undefined at $θ$ .

$\csc (θ) = Undefined$

Undefined

This is the solution to each trig value.

Undefined

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