Trigonometry Examples

$\csc (θ) = 5$ with $\frac{π}{2} < θ < π$

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.

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

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.

$Adjacent = - \sqrt{{(5)}^{2} - {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Negate $\sqrt{{(5)}^{2} - {(1)}^{2}}$ .

Adjacent $= - \sqrt{{(5)}^{2} - {(1)}^{2}}$

Step 4.2

Raise $5$ to the power of $2$ .

Adjacent $= - \sqrt{25 - {(1)}^{2}}$

Step 4.3

One to any power is one.

Adjacent $= - \sqrt{25 - 1 \cdot 1}$

Step 4.4

Multiply $- 1$ by $1$ .

Adjacent $= - \sqrt{25 - 1}$

Step 4.5

Subtract $1$ from $25$ .

Adjacent $= - \sqrt{24}$

Step 4.6

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 4.6.1

Factor $4$ out of $24$ .

Adjacent $= - \sqrt{4 (6)}$

Step 4.6.2

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

Adjacent $= - \sqrt{2^{2} \cdot 6}$

Step 4.7

Pull terms out from under the radical.

Adjacent $= - (2 \sqrt{6})$

Step 4.8

Multiply $2$ by $- 1$ .

Adjacent $= - 2 \sqrt{6}$

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.

$\sin (θ) = \frac{1}{5}$

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.

$\cos (θ) = \frac{- 2 \sqrt{6}}{5}$

Step 6.3

Move the negative in front of the fraction.

$\cos (θ) = - \frac{2 \sqrt{6}}{5}$

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.

$\tan (θ) = \frac{1}{- 2 \sqrt{6}}$

Step 7.3

Simplify the value of $\tan (θ)$ .

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Step 7.3.1

Move the negative in front of the fraction.

$\tan (θ) = - \frac{1}{2 \sqrt{6}}$

Step 7.3.2

Multiply $\frac{1}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

Multiply $\frac{1}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Step 7.3.3.2

Move $\sqrt{6}$ .

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

Step 7.3.3.3

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 7.3.3.4

Raise $\sqrt{6}$ to the power of $1$ .

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

Step 7.3.3.5

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

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

Step 7.3.3.6

Add $1$ and $1$ .

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

Step 7.3.3.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 7.3.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 7.3.3.7.2

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

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

Step 7.3.3.7.3

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

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

Step 7.3.3.7.4

Cancel the common factor of $2$ .

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Step 7.3.3.7.4.1

Cancel the common factor.

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

Step 7.3.3.7.4.2

Rewrite the expression.

$\tan (θ) = - \frac{\sqrt{6}}{2 \cdot 6}$

Step 7.3.3.7.5

Evaluate the exponent.

$\tan (θ) = - \frac{\sqrt{6}}{2 \cdot 6}$

Step 7.3.4

Multiply $2$ by $6$ .

$\tan (θ) = - \frac{\sqrt{6}}{12}$

Step 8

Find the value of cotangent.

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Step 8.1

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

$\cot (θ) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

$\cot (θ) = \frac{- 2 \sqrt{6}}{1}$

Step 8.3

Divide $- 2 \sqrt{6}$ by $1$ .

$\cot (θ) = - 2 \sqrt{6}$

Step 9

Find the value of secant.

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Step 9.1

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

$\sec (θ) = \frac{hyp}{adj}$

Step 9.2

Substitute in the known values.

$\sec (θ) = \frac{5}{- 2 \sqrt{6}}$

Step 9.3

Simplify the value of $\sec (θ)$ .

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Step 9.3.1

Move the negative in front of the fraction.

$\sec (θ) = - \frac{5}{2 \sqrt{6}}$

Step 9.3.2

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sec (θ) = - (\frac{5}{2 \sqrt{6}} \cdot \frac{\sqrt{6}}{\sqrt{6}})$

Step 9.3.3

Combine and simplify the denominator.

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Step 9.3.3.1

Multiply $\frac{5}{2 \sqrt{6}}$ by $\frac{\sqrt{6}}{\sqrt{6}}$ .

$\sec (θ) = - \frac{5 \sqrt{6}}{2 \sqrt{6} \sqrt{6}}$

Step 9.3.3.2

Move $\sqrt{6}$ .

$\sec (θ) = - \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 9.3.3.3

Raise $\sqrt{6}$ to the power of $1$ .

$\sec (θ) = - \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 9.3.3.4

Raise $\sqrt{6}$ to the power of $1$ .

$\sec (θ) = - \frac{5 \sqrt{6}}{2 (\sqrt{6} \sqrt{6})}$

Step 9.3.3.5

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

$\sec (θ) = - \frac{5 \sqrt{6}}{2 {\sqrt{6}}^{1 + 1}}$

Step 9.3.3.6

Add $1$ and $1$ .

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

Step 9.3.3.7

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Step 9.3.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

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

Step 9.3.3.7.2

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

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

Step 9.3.3.7.3

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

$\sec (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 9.3.3.7.4

Cancel the common factor of $2$ .

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Step 9.3.3.7.4.1

Cancel the common factor.

$\sec (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6^{\frac{2}{2}}}$

Step 9.3.3.7.4.2

Rewrite the expression.

$\sec (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6}$

Step 9.3.3.7.5

Evaluate the exponent.

$\sec (θ) = - \frac{5 \sqrt{6}}{2 \cdot 6}$

Step 9.3.4

Multiply $2$ by $6$ .

$\sec (θ) = - \frac{5 \sqrt{6}}{12}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{1}{5}$

$\cos (θ) = - \frac{2 \sqrt{6}}{5}$

$\tan (θ) = - \frac{\sqrt{6}}{12}$

$\cot (θ) = - 2 \sqrt{6}$

$\sec (θ) = - \frac{5 \sqrt{6}}{12}$

$\csc (θ) = 5$