Find the Other Trig Values in Quadrant II sin(theta)=1/3

$\sin (θ) = \frac{1}{3}$

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.

$Adjacent = - \sqrt{{(3)}^{2} - {(1)}^{2}}$

Simplify inside the radical.

Negate $\sqrt{{(3)}^{2} - {(1)}^{2}}$ .

Adjacent $= - \sqrt{{(3)}^{2} - {(1)}^{2}}$

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

Adjacent $= - \sqrt{9 - {(1)}^{2}}$

One to any power is one.

Adjacent $= - \sqrt{9 - 1 \cdot 1}$

Multiply $- 1$ by $1$ .

Adjacent $= - \sqrt{9 - 1}$

Subtract $1$ from $9$ .

Adjacent $= - \sqrt{8}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

Adjacent $= - (2 \sqrt{2})$

Multiply $2$ by $- 1$ .

Adjacent $= - 2 \sqrt{2}$

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.

$\cos (θ) = \frac{- 2 \sqrt{2}}{3}$

Move the negative in front of the fraction.

$\cos (θ) = - \frac{2 \sqrt{2}}{3}$

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.

$\tan (θ) = \frac{1}{- 2 \sqrt{2}}$

Simplify the value of $\tan (θ)$ .

Move the negative in front of the fraction.

$\tan (θ) = - \frac{1}{(2) \sqrt{2}}$

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (θ) = - (\frac{1}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Combine and simplify the denominator.

Multiply $\frac{1}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\tan (θ) = - \frac{\sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 {\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\tan (θ) = - \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\tan (θ) = - \frac{\sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\tan (θ) = - \frac{\sqrt{2}}{2 \cdot 2}$

Evaluate the exponent.

$\tan (θ) = - \frac{\sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\tan (θ) = - \frac{\sqrt{2}}{4}$

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.

$\cot (θ) = \frac{- 2 \sqrt{2}}{1}$

Divide $- 2 \sqrt{2}$ by $1$ .

$\cot (θ) = - 2 \sqrt{2}$

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.

$\sec (θ) = \frac{3}{- 2 \sqrt{2}}$

Simplify the value of $\sec (θ)$ .

Move the negative in front of the fraction.

$\sec (θ) = - \frac{3}{(2) \sqrt{2}}$

Multiply $\frac{3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (θ) = - (\frac{3}{2 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Combine and simplify the denominator.

Multiply $\frac{3}{2 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 \sqrt{2} \sqrt{2}}$

Move $\sqrt{2}$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Raise $\sqrt{2}$ to the power of $1$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 (\sqrt{2} \sqrt{2})}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (θ) = - \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{1 + 1}}$

Add $1$ and $1$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 {\sqrt{2}}^{2}}$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 {(2^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\sec (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2^{\frac{2}{2}}}$

Rewrite the expression.

$\sec (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2}$

Evaluate the exponent.

$\sec (θ) = - \frac{3 \sqrt{2}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$\sec (θ) = - \frac{3 \sqrt{2}}{4}$

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.

$\csc (θ) = \frac{3}{1}$

Divide $3$ by $1$ .

$\csc (θ) = 3$

This is the solution to each trig value.

$\sin (θ) = \frac{1}{3}$

$\cos (θ) = - \frac{2 \sqrt{2}}{3}$

$\tan (θ) = - \frac{\sqrt{2}}{4}$

$\cot (θ) = - 2 \sqrt{2}$

$\sec (θ) = - \frac{3 \sqrt{2}}{4}$

$\csc (θ) = 3$

Find the Other Trig Values in Quadrant II sin(theta)=1/3

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