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

$\sin (θ) = 0$

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.

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

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.

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

Simplify inside the radical.

Negate $\sqrt{{(1)}^{2} - {(0)}^{2}}$ .

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

One to any power is one.

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

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

Adjacent $= - \sqrt{1 - 0}$

Multiply $- 1$ by $0$ .

Adjacent $= - \sqrt{1 + 0}$

Add $1$ and $0$ .

Adjacent $= - \sqrt{1}$

Any root of $1$ is $1$ .

Adjacent $= - 1 \cdot 1$

Multiply $- 1$ by $1$ .

Adjacent $= - 1$

Find the value of cosine.

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 tangent.

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

$\tan (θ) = \frac{opp}{adj}$

Substitute in the known values.

$\tan (θ) = \frac{0}{- 1}$

Divide $0$ by $- 1$ .

$\tan (θ) = 0$

Find the value of cotangent.

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.

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.

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

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

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