Find the Other Trig Values in Quadrant II csc(theta) = square root of 2

$\csc (θ) = \sqrt{2}$

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

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

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.

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

Simplify inside the radical.

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

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

One to any power is one.

Adjacent $= - \sqrt{2 - 1 \cdot 1}$

Multiply $- 1$ by $1$ .

Adjacent $= - \sqrt{2 - 1}$

Subtract $1$ from $2$ .

Adjacent $= - \sqrt{1}$

Any root of $1$ is $1$ .

Adjacent $= - 1 \cdot 1$

Multiply $- 1$ by $1$ .

Adjacent $= - 1$

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.

$\sin (θ) = \frac{1}{\sqrt{2}}$

Simplify the value of $\sin (θ)$ .

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

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

Combine and simplify the denominator.

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

$\sin (θ) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

$\sin (θ) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

$\sin (θ) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

$\sin (θ) = \frac{\sqrt{2}}{2}$

Evaluate the exponent.

$\sin (θ) = \frac{\sqrt{2}}{2}$

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.

$\cos (θ) = \frac{- 1}{\sqrt{2}}$

Simplify the value of $\cos (θ)$ .

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Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{\sqrt{2}}$

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

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

Combine and simplify the denominator.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{2}}{2}$

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{2}}{2}$

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.

$\tan (θ) = \frac{1}{- 1}$

Divide $1$ by $- 1$ .

$\tan (θ) = - 1$

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.

$\cot (θ) = \frac{- 1}{1}$

Divide $- 1$ by $1$ .

$\cot (θ) = - 1$

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.

$\sec (θ) = \frac{\sqrt{2}}{- 1}$

Simplify the value of $\sec (θ)$ .

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Move the negative one from the denominator of $\frac{\sqrt{2}}{- 1}$ .

$\sec (θ) = - 1 \cdot \sqrt{2}$

Rewrite $- 1 \cdot \sqrt{2}$ as $- \sqrt{2}$ .

$\sec (θ) = - \sqrt{2}$

This is the solution to each trig value.

$\sin (θ) = \frac{\sqrt{2}}{2}$

$\cos (θ) = - \frac{\sqrt{2}}{2}$

$\tan (θ) = - 1$

$\cot (θ) = - 1$

$\sec (θ) = - \sqrt{2}$

$\csc (θ) = \sqrt{2}$

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