Precalculus Examples

$\csc (x) = \sqrt{3}$

Step 1

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.

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

Step 2

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.

$Adjacent = - \sqrt{{hypotenuse}^{2} - {opposite}^{2}}$

Step 3

Replace the known values in the equation.

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

Step 4

Simplify inside the radical.

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Step 4.1

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

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

Step 4.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 4.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

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

Step 4.2.4.2

Rewrite the expression.

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

Step 4.2.5

Evaluate the exponent.

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

Step 4.3

One to any power is one.

Adjacent $= - \sqrt{3 - 1 \cdot 1}$

Step 4.4

Multiply $- 1$ by $1$ .

Adjacent $= - \sqrt{3 - 1}$

Step 4.5

Subtract $1$ from $3$ .

Adjacent $= - \sqrt{2}$

Step 5

Find the value of sine.

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Step 5.1

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

$\sin (x) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (x) = \frac{1}{\sqrt{3}}$

Step 5.3

Simplify the value of $\sin (x)$ .

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Step 5.3.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sin (x) = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.3.2

Combine and simplify the denominator.

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Step 5.3.2.1

Multiply $\frac{1}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sin (x) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.3.2.2

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (x) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.3.2.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sin (x) = \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.3.2.4

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

$\sin (x) = \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.3.2.5

Add $1$ and $1$ .

$\sin (x) = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.3.2.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\sin (x) = \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.3.2.6.2

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

$\sin (x) = \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.6.3

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

$\sin (x) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.3.2.6.4

Cancel the common factor of $2$ .

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Step 5.3.2.6.4.1

Cancel the common factor.

$\sin (x) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.3.2.6.4.2

Rewrite the expression.

$\sin (x) = \frac{\sqrt{3}}{3}$

Step 5.3.2.6.5

Evaluate the exponent.

$\sin (x) = \frac{\sqrt{3}}{3}$

Step 6

Find the value of cosine.

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Step 6.1

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

$\cos (x) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (x) = \frac{- \sqrt{2}}{\sqrt{3}}$

Step 6.3

Simplify the value of $\cos (x)$ .

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Step 6.3.1

Move the negative in front of the fraction.

$\cos (x) = - \frac{\sqrt{2}}{\sqrt{3}}$

Step 6.3.2

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

$\cos (x) = - (\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

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

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.3.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.3.3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 6.3.3.4

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

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 6.3.3.5

Add $1$ and $1$ .

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 6.3.3.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 6.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

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

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

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

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.3.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.3.6.4.1

Cancel the common factor.

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{3}$

Step 6.3.3.6.5

Evaluate the exponent.

$\cos (x) = - \frac{\sqrt{2} \sqrt{3}}{3}$

Step 6.3.4

Simplify the numerator.

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Step 6.3.4.1

Combine using the product rule for radicals.

$\cos (x) = - \frac{\sqrt{2 \cdot 3}}{3}$

Step 6.3.4.2

Multiply $2$ by $3$ .

$\cos (x) = - \frac{\sqrt{6}}{3}$

Step 7

Find the value of tangent.

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Step 7.1

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

$\tan (x) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

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

Step 7.3

Simplify the value of $\tan (x)$ .

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Step 7.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 7.3.1.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 7.3.1.2

Move the negative in front of the fraction.

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

Step 7.3.2

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

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

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

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

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

Step 7.3.3.2

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

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

Step 7.3.3.3

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

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

Step 7.3.3.4

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

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

Step 7.3.3.5

Add $1$ and $1$ .

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

Step 7.3.3.6

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

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Step 7.3.3.6.1

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

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

Step 7.3.3.6.2

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

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

Step 7.3.3.6.3

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

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

Step 7.3.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.3.6.4.1

Cancel the common factor.

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

Step 7.3.3.6.4.2

Rewrite the expression.

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

Step 7.3.3.6.5

Evaluate the exponent.

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

Step 8

Find the value of cotangent.

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Step 8.1

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

$\cot (x) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

$\cot (x) = \frac{- \sqrt{2}}{1}$

Step 8.3

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

$\cot (x) = - \sqrt{2}$

Step 9

Find the value of secant.

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Step 9.1

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

$\sec (x) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

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

Step 9.3

Simplify the value of $\sec (x)$ .

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Step 9.3.1

Move the negative in front of the fraction.

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

Step 9.3.2

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

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

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

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

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

Step 9.3.3.2

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

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

Step 9.3.3.3

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

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

Step 9.3.3.4

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

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

Step 9.3.3.5

Add $1$ and $1$ .

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

Step 9.3.3.6

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

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Step 9.3.3.6.1

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

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

Step 9.3.3.6.2

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

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

Step 9.3.3.6.3

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

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

Step 9.3.3.6.4

Cancel the common factor of $2$ .

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Step 9.3.3.6.4.1

Cancel the common factor.

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

Step 9.3.3.6.4.2

Rewrite the expression.

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

Step 9.3.3.6.5

Evaluate the exponent.

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

Step 9.3.4

Simplify the numerator.

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Step 9.3.4.1

Combine using the product rule for radicals.

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

Step 9.3.4.2

Multiply $3$ by $2$ .

$\sec (x) = - \frac{\sqrt{6}}{2}$

Step 10

This is the solution to each trig value.

$\sin (x) = \frac{\sqrt{3}}{3}$

$\cos (x) = - \frac{\sqrt{6}}{3}$

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

$\cot (x) = - \sqrt{2}$

$\sec (x) = - \frac{\sqrt{6}}{2}$

$\csc (x) = \sqrt{3}$