Trigonometry Examples

csc (θ) = 5

$\csc (θ) = 5$ with

\frac{π}{2} < θ < π

$\frac{π}{2} < θ < π$

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}

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

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}}

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

Step 3

Replace the known values in the equation.

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

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

Step 4

Simplify inside the radical.

Tap for more steps...

Step 4.1

Negate

\sqrt{{(5)}^{2} - {(1)}^{2}}

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

Adjacent

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

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

Step 4.2

Raise

5

$5$ to the power of

2

$2$ .

Adjacent

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

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

Step 4.3

One to any power is one.

Adjacent

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

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

Step 4.4

Multiply

- 1

$- 1$ by

1

$1$ .

Adjacent

= - \sqrt{25 - 1}

$= - \sqrt{25 - 1}$

Step 4.5

Subtract

1

$1$ from

25

$25$ .

Adjacent

= - \sqrt{24}

$= - \sqrt{24}$

Step 4.6

Rewrite

24

$24$ as

2^{2} \cdot 6

$2^{2} \cdot 6$ .

Tap for more steps...

Step 4.6.1

Factor

4

$4$ out of

24

$24$ .

Adjacent

= - \sqrt{4 (6)}

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

Step 4.6.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

Adjacent

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

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

Adjacent

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

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

Step 4.7

Pull terms out from under the radical.

Adjacent

= - (2 \sqrt{6})

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

Step 4.8

Multiply

2

$2$ by

- 1

$- 1$ .

Adjacent

= - 2 \sqrt{6}

$= - 2 \sqrt{6}$

Adjacent

= - 2 \sqrt{6}

$= - 2 \sqrt{6}$

Step 5

Find the value of sine.

Tap for more steps...

Step 5.1

Use the definition of sine to find the value of

sin (θ)

$\sin (θ)$ .

sin (θ) = \frac{opp}{hyp}

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

sin (θ) = \frac{1}{5}

$\sin (θ) = \frac{1}{5}$

sin (θ) = \frac{1}{5}

$\sin (θ) = \frac{1}{5}$

Step 6

Find the value of cosine.

Tap for more steps...

Step 6.1

Use the definition of cosine to find the value of

cos (θ)

$\cos (θ)$ .

cos (θ) = \frac{adj}{hyp}

$\cos (θ) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

cos (θ) = \frac{- 2 \sqrt{6}}{5}

$\cos (θ) = \frac{- 2 \sqrt{6}}{5}$

Step 6.3

Move the negative in front of the fraction.

cos (θ) = - \frac{2 \sqrt{6}}{5}

$\cos (θ) = - \frac{2 \sqrt{6}}{5}$

cos (θ) = - \frac{2 \sqrt{6}}{5}

$\cos (θ) = - \frac{2 \sqrt{6}}{5}$

Step 7

Find the value of tangent.

Tap for more steps...

Step 7.1

Use the definition of tangent to find the value of

tan (θ)

$\tan (θ)$ .

tan (θ) = \frac{opp}{adj}

$\tan (θ) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

tan (θ) = \frac{1}{- 2 \sqrt{6}}

$\tan (θ) = \frac{1}{- 2 \sqrt{6}}$

Step 7.3

Simplify the value of

tan (θ)

$\tan (θ)$ .

Tap for more steps...

Step 7.3.1

Move the negative in front of the fraction.

tan (θ) = - \frac{1}{2 \sqrt{6}}

$\tan (θ) = - \frac{1}{2 \sqrt{6}}$

Step 7.3.2

Multiply

\frac{1}{2 \sqrt{6}}

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

\frac{\sqrt{6}}{\sqrt{6}}

$\frac{\sqrt{6}}{\sqrt{6}}$ .

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

Tap for more steps...

Step 7.3.3.1

Multiply

\frac{1}{2 \sqrt{6}}

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

\frac{\sqrt{6}}{\sqrt{6}}

$\frac{\sqrt{6}}{\sqrt{6}}$ .

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

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

Step 7.3.3.2

Move

\sqrt{6}

$\sqrt{6}$ .

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

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

Step 7.3.3.3

Raise

\sqrt{6}

$\sqrt{6}$ to the power of

1

$1$ .

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

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

Step 7.3.3.4

Raise

\sqrt{6}

$\sqrt{6}$ to the power of

1

$1$ .

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

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

Step 7.3.3.5

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

Step 7.3.3.6

Add

1

$1$ and

1

$1$ .

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

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

Step 7.3.3.7

Rewrite

{\sqrt{6}}^{2}

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

6

$6$ .

Tap for more steps...

Step 7.3.3.7.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{6}

$\sqrt{6}$ as

6^{\frac{1}{2}}

$6^{\frac{1}{2}}$ .

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

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 7.3.3.7.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 7.3.3.7.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 7.3.3.7.4.1

Cancel the common factor.

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

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

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.

Tap for more steps...

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.

Tap for more steps...

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

$\sec (θ)$ .

Tap for more steps...

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.

Tap for more steps...

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

$6$ .

Tap for more steps...

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

$2$ .

Tap for more steps...

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.

$\sin (θ) = \frac{1}{5}$

$\cos (θ) = - \frac{2 \sqrt{6}}{5}$

$\tan (θ) = - \frac{\sqrt{6}}{12}$

$\cot (θ) = - 2 \sqrt{6}$

$\sec (θ) = - \frac{5 \sqrt{6}}{12}$

$\csc (θ) = 5$