Trigonometry Examples

$\cot (θ) = \frac{\sqrt{5}}{6}$

Step 1

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

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

Step 2

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.

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

Step 3

Replace the known values in the equation.

$Hypotenuse = \sqrt{{(- 6)}^{2} + {(- \sqrt{5})}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise $- 6$ to the power of $2$ .

Hypotenuse $= \sqrt{36 + {(- \sqrt{5})}^{2}}$

Step 4.2

Apply the product rule to $- \sqrt{5}$ .

Hypotenuse $= \sqrt{36 + {(- 1)}^{2} {\sqrt{5}}^{2}}$

Step 4.3

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

Hypotenuse $= \sqrt{36 + 1 {\sqrt{5}}^{2}}$

Step 4.4

Multiply ${\sqrt{5}}^{2}$ by $1$ .

Hypotenuse $= \sqrt{36 + {\sqrt{5}}^{2}}$

Step 4.5

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

Hypotenuse $= \sqrt{36 + {(5^{\frac{1}{2}})}^{2}}$

Step 4.5.2

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

Hypotenuse $= \sqrt{36 + 5^{\frac{1}{2} \cdot 2}}$

Step 4.5.3

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

Hypotenuse $= \sqrt{36 + 5^{\frac{2}{2}}}$

Step 4.5.4

Cancel the common factor of $2$ .

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Step 4.5.4.1

Cancel the common factor.

Hypotenuse $= \sqrt{36 + 5^{\frac{2}{2}}}$

Step 4.5.4.2

Rewrite the expression.

Hypotenuse $= \sqrt{36 + 5}$

Step 4.5.5

Evaluate the exponent.

Hypotenuse $= \sqrt{36 + 5}$

Step 4.6

Add $36$ and $5$ .

Hypotenuse $= \sqrt{41}$

Step 5

Find the value of sine.

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Step 5.1

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

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

$\sin (θ) = \frac{- 6}{\sqrt{41}}$

Step 5.3

Simplify the value of $\sin (θ)$ .

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Step 5.3.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{6}{\sqrt{41}}$

Step 5.3.2

Multiply $\frac{6}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\sin (θ) = - (\frac{6}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}})$

Step 5.3.3

Combine and simplify the denominator.

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Step 5.3.3.1

Multiply $\frac{6}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\sin (θ) = - \frac{6 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 5.3.3.2

Raise $\sqrt{41}$ to the power of $1$ .

$\sin (θ) = - \frac{6 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 5.3.3.3

Raise $\sqrt{41}$ to the power of $1$ .

$\sin (θ) = - \frac{6 \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 5.3.3.4

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

$\sin (θ) = - \frac{6 \sqrt{41}}{{\sqrt{41}}^{1 + 1}}$

Step 5.3.3.5

Add $1$ and $1$ .

$\sin (θ) = - \frac{6 \sqrt{41}}{{\sqrt{41}}^{2}}$

Step 5.3.3.6

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

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Step 5.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{6 \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}$

Step 5.3.3.6.2

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

$\sin (θ) = - \frac{6 \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}$

Step 5.3.3.6.3

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

$\sin (θ) = - \frac{6 \sqrt{41}}{41^{\frac{2}{2}}}$

Step 5.3.3.6.4

Cancel the common factor of $2$ .

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Step 5.3.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{6 \sqrt{41}}{41^{\frac{2}{2}}}$

Step 5.3.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{6 \sqrt{41}}{41}$

Step 5.3.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{6 \sqrt{41}}{41}$

Step 6

Find the value of cosine.

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Step 6.1

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

$\cos (θ) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (θ) = \frac{- \sqrt{5}}{\sqrt{41}}$

Step 6.3

Simplify the value of $\cos (θ)$ .

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Step 6.3.1

Move the negative in front of the fraction.

$\cos (θ) = - \frac{\sqrt{5}}{\sqrt{41}}$

Step 6.3.2

Multiply $\frac{\sqrt{5}}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\cos (θ) = - (\frac{\sqrt{5}}{\sqrt{41}} \cdot \frac{\sqrt{41}}{\sqrt{41}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply $\frac{\sqrt{5}}{\sqrt{41}}$ by $\frac{\sqrt{41}}{\sqrt{41}}$ .

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 6.3.3.2

Raise $\sqrt{41}$ to the power of $1$ .

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 6.3.3.3

Raise $\sqrt{41}$ to the power of $1$ .

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{\sqrt{41} \sqrt{41}}$

Step 6.3.3.4

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

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{{\sqrt{41}}^{1 + 1}}$

Step 6.3.3.5

Add $1$ and $1$ .

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{{\sqrt{41}}^{2}}$

Step 6.3.3.6

Rewrite ${\sqrt{41}}^{2}$ as $41$ .

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Step 6.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{41}$ as $41^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{{(41^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

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

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{41^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

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

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{41^{\frac{2}{2}}}$

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.

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{41^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{41}$

Step 6.3.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{5} \sqrt{41}}{41}$

Step 6.3.4

Simplify the numerator.

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Step 6.3.4.1

Combine using the product rule for radicals.

$\cos (θ) = - \frac{\sqrt{5 \cdot 41}}{41}$

Step 6.3.4.2

Multiply $5$ by $41$ .

$\cos (θ) = - \frac{\sqrt{205}}{41}$

Step 7

Find the value of tangent.

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Step 7.1

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

$\tan (θ) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

$\tan (θ) = \frac{- 6}{- \sqrt{5}}$

Step 7.3

Simplify the value of $\tan (θ)$ .

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Step 7.3.1

Dividing two negative values results in a positive value.

$\tan (θ) = \frac{6}{\sqrt{5}}$

Step 7.3.2

Multiply $\frac{6}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\tan (θ) = \frac{6}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

Multiply $\frac{6}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\tan (θ) = \frac{6 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\tan (θ) = \frac{6 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\tan (θ) = \frac{6 \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 7.3.3.4

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

$\tan (θ) = \frac{6 \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 7.3.3.5

Add $1$ and $1$ .

$\tan (θ) = \frac{6 \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 7.3.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 7.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\tan (θ) = \frac{6 \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

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

$\tan (θ) = \frac{6 \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

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

$\tan (θ) = \frac{6 \sqrt{5}}{5^{\frac{2}{2}}}$

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.

$\tan (θ) = \frac{6 \sqrt{5}}{5^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\tan (θ) = \frac{6 \sqrt{5}}{5}$

Step 7.3.3.6.5

Evaluate the exponent.

$\tan (θ) = \frac{6 \sqrt{5}}{5}$

Step 8

Find the value of secant.

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Step 8.1

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

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ) = \frac{\sqrt{41}}{- \sqrt{5}}$

Step 8.3

Simplify the value of $\sec (θ)$ .

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Step 8.3.1

Move the negative in front of the fraction.

$\sec (θ) = - \frac{\sqrt{41}}{\sqrt{5}}$

Step 8.3.2

Multiply $\frac{\sqrt{41}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sec (θ) = - (\frac{\sqrt{41}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply $\frac{\sqrt{41}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.3.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.3.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.3.3.4

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

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 8.3.3.5

Add $1$ and $1$ .

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 8.3.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 8.3.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 8.3.3.6.2

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

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 8.3.3.6.3

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

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.3.3.6.4

Cancel the common factor of $2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.3.3.6.4.2

Rewrite the expression.

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{5}$

Step 8.3.3.6.5

Evaluate the exponent.

$\sec (θ) = - \frac{\sqrt{41} \sqrt{5}}{5}$

Step 8.3.4

Simplify the numerator.

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Step 8.3.4.1

Combine using the product rule for radicals.

$\sec (θ) = - \frac{\sqrt{41 \cdot 5}}{5}$

Step 8.3.4.2

Multiply $41$ by $5$ .

$\sec (θ) = - \frac{\sqrt{205}}{5}$

Step 9

Find the value of cosecant.

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Step 9.1

Use the definition of cosecant to find the value of $\csc (θ)$ .

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

$\csc (θ) = \frac{\sqrt{41}}{- 6}$

Step 9.3

Move the negative in front of the fraction.

$\csc (θ) = - \frac{\sqrt{41}}{6}$

Step 10

This is the solution to each trig value.

$\sin (θ) = - \frac{6 \sqrt{41}}{41}$

$\cos (θ) = - \frac{\sqrt{205}}{41}$

$\tan (θ) = \frac{6 \sqrt{5}}{5}$

$\cot (θ) = \frac{\sqrt{5}}{6}$

$\sec (θ) = - \frac{\sqrt{205}}{5}$

$\csc (θ) = - \frac{\sqrt{41}}{6}$