Trigonometry Examples

\tan (θ) = 6

$\tan (θ) = 6$

Step 1

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

\tan (θ) = \frac{opposite}{adjacent}

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

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}}

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

Step 3

Replace the known values in the equation.

Hypotenuse = \sqrt{{(6)}^{2} + {(1)}^{2}}

$Hypotenuse = \sqrt{{(6)}^{2} + {(1)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise

6

$6$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{36 + {(1)}^{2}}

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

Step 4.2

One to any power is one.

Hypotenuse

= \sqrt{36 + 1}

$= \sqrt{36 + 1}$

Step 4.3

Add

36

$36$ and

1

$1$ .

Hypotenuse

= \sqrt{37}

$= \sqrt{37}$

Hypotenuse

= \sqrt{37}

$= \sqrt{37}$

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)

$\sin (θ)$ .

\sin (θ) = \frac{opp}{hyp}

$\sin (θ) = \frac{opp}{hyp}$

Step 5.2

Substitute in the known values.

\sin (θ) = \frac{6}{\sqrt{37}}

$\sin (θ) = \frac{6}{\sqrt{37}}$

Step 5.3

Simplify the value of

\sin (θ)

$\sin (θ)$ .

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Step 5.3.1

Multiply

\frac{6}{\sqrt{37}}

$\frac{6}{\sqrt{37}}$ by

\frac{\sqrt{37}}{\sqrt{37}}

$\frac{\sqrt{37}}{\sqrt{37}}$ .

\sin (θ) = \frac{6}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}}

$\sin (θ) = \frac{6}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}}$

Step 5.3.2

Combine and simplify the denominator.

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Step 5.3.2.1

Multiply

\frac{6}{\sqrt{37}}

$\frac{6}{\sqrt{37}}$ by

\frac{\sqrt{37}}{\sqrt{37}}

$\frac{\sqrt{37}}{\sqrt{37}}$ .

\sin (θ) = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}

$\sin (θ) = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 5.3.2.2

Raise

\sqrt{37}

$\sqrt{37}$ to the power of

1

$1$ .

\sin (θ) = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}

$\sin (θ) = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 5.3.2.3

Raise

\sqrt{37}

$\sqrt{37}$ to the power of

1

$1$ .

\sin (θ) = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}

$\sin (θ) = \frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 5.3.2.4

Use the power rule

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

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

\sin (θ) = \frac{6 \sqrt{37}}{{\sqrt{37}}^{1 + 1}}

$\sin (θ) = \frac{6 \sqrt{37}}{{\sqrt{37}}^{1 + 1}}$

Step 5.3.2.5

Add

1

$1$ and

1

$1$ .

\sin (θ) = \frac{6 \sqrt{37}}{{\sqrt{37}}^{2}}

$\sin (θ) = \frac{6 \sqrt{37}}{{\sqrt{37}}^{2}}$

Step 5.3.2.6

Rewrite

{\sqrt{37}}^{2}

${\sqrt{37}}^{2}$ as

37

$37$ .

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Step 5.3.2.6.1

Use

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

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

\sqrt{37}

$\sqrt{37}$ as

37^{\frac{1}{2}}

$37^{\frac{1}{2}}$ .

\sin (θ) = \frac{6 \sqrt{37}}{{(37^{\frac{1}{2}})}^{2}}

$\sin (θ) = \frac{6 \sqrt{37}}{{(37^{\frac{1}{2}})}^{2}}$

Step 5.3.2.6.2

Apply the power rule and multiply exponents,

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

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

\sin (θ) = \frac{6 \sqrt{37}}{37^{\frac{1}{2} \cdot 2}}

$\sin (θ) = \frac{6 \sqrt{37}}{37^{\frac{1}{2} \cdot 2}}$

Step 5.3.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sin (θ) = \frac{6 \sqrt{37}}{37^{\frac{2}{2}}}

$\sin (θ) = \frac{6 \sqrt{37}}{37^{\frac{2}{2}}}$

Step 5.3.2.6.4

Cancel the common factor of

2

$2$ .

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Step 5.3.2.6.4.1

Cancel the common factor.

\sin (θ) = \frac{6 \sqrt{37}}{37^{\frac{2}{2}}}

$\sin (θ) = \frac{6 \sqrt{37}}{37^{\frac{2}{2}}}$

Step 5.3.2.6.4.2

Rewrite the expression.

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

Step 5.3.2.6.5

Evaluate the exponent.

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

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)

$\cos (θ)$ .

\cos (θ) = \frac{adj}{hyp}

$\cos (θ) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

\cos (θ) = \frac{1}{\sqrt{37}}

$\cos (θ) = \frac{1}{\sqrt{37}}$

Step 6.3

Simplify the value of

\cos (θ)

$\cos (θ)$ .

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Step 6.3.1

Multiply

\frac{1}{\sqrt{37}}

$\frac{1}{\sqrt{37}}$ by

\frac{\sqrt{37}}{\sqrt{37}}

$\frac{\sqrt{37}}{\sqrt{37}}$ .

\cos (θ) = \frac{1}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}}

$\cos (θ) = \frac{1}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply

\frac{1}{\sqrt{37}}

$\frac{1}{\sqrt{37}}$ by

\frac{\sqrt{37}}{\sqrt{37}}

$\frac{\sqrt{37}}{\sqrt{37}}$ .

\cos (θ) = \frac{\sqrt{37}}{\sqrt{37} \sqrt{37}}

$\cos (θ) = \frac{\sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 6.3.2.2

Raise

\sqrt{37}

$\sqrt{37}$ to the power of

1

$1$ .

\cos (θ) = \frac{\sqrt{37}}{\sqrt{37} \sqrt{37}}

$\cos (θ) = \frac{\sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 6.3.2.3

Raise

\sqrt{37}

$\sqrt{37}$ to the power of

1

$1$ .

\cos (θ) = \frac{\sqrt{37}}{\sqrt{37} \sqrt{37}}

$\cos (θ) = \frac{\sqrt{37}}{\sqrt{37} \sqrt{37}}$

Step 6.3.2.4

Use the power rule

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

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

\cos (θ) = \frac{\sqrt{37}}{{\sqrt{37}}^{1 + 1}}

$\cos (θ) = \frac{\sqrt{37}}{{\sqrt{37}}^{1 + 1}}$

Step 6.3.2.5

Add

1

$1$ and

1

$1$ .

\cos (θ) = \frac{\sqrt{37}}{{\sqrt{37}}^{2}}

$\cos (θ) = \frac{\sqrt{37}}{{\sqrt{37}}^{2}}$

Step 6.3.2.6

Rewrite

{\sqrt{37}}^{2}

${\sqrt{37}}^{2}$ as

37

$37$ .

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Step 6.3.2.6.1

Use

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

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

\sqrt{37}

$\sqrt{37}$ as

37^{\frac{1}{2}}

$37^{\frac{1}{2}}$ .

\cos (θ) = \frac{\sqrt{37}}{{(37^{\frac{1}{2}})}^{2}}

$\cos (θ) = \frac{\sqrt{37}}{{(37^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

Apply the power rule and multiply exponents,

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

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

\cos (θ) = \frac{\sqrt{37}}{37^{\frac{1}{2} \cdot 2}}

$\cos (θ) = \frac{\sqrt{37}}{37^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\cos (θ) = \frac{\sqrt{37}}{37^{\frac{2}{2}}}

$\cos (θ) = \frac{\sqrt{37}}{37^{\frac{2}{2}}}$

Step 6.3.2.6.4

Cancel the common factor of

2

$2$ .

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Step 6.3.2.6.4.1

Cancel the common factor.

\cos (θ) = \frac{\sqrt{37}}{37^{\frac{2}{2}}}

$\cos (θ) = \frac{\sqrt{37}}{37^{\frac{2}{2}}}$

Step 6.3.2.6.4.2

Rewrite the expression.

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

Step 6.3.2.6.5

Evaluate the exponent.

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

Step 7

Find the value of cotangent.

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Step 7.1

Use the definition of cotangent to find the value of

\cot (θ)

$\cot (θ)$ .

\cot (θ) = \frac{adj}{opp}

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

\cot (θ) = \frac{1}{6}

$\cot (θ) = \frac{1}{6}$

\cot (θ) = \frac{1}{6}

$\cot (θ) = \frac{1}{6}$

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)

$\sec (θ)$ .

\sec (θ) = \frac{hyp}{adj}

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

\sec (θ) = \frac{\sqrt{37}}{1}

$\sec (θ) = \frac{\sqrt{37}}{1}$

Step 8.3

Divide

\sqrt{37}

$\sqrt{37}$ by

1

$1$ .

\sec (θ) = \sqrt{37}

$\sec (θ) = \sqrt{37}$

\sec (θ) = \sqrt{37}

$\sec (θ) = \sqrt{37}$

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)

$\csc (θ)$ .

\csc (θ) = \frac{hyp}{opp}

$\csc (θ) = \frac{hyp}{opp}$

Step 9.2

Substitute in the known values.

\csc (θ) = \frac{\sqrt{37}}{6}

$\csc (θ) = \frac{\sqrt{37}}{6}$

\csc (θ) = \frac{\sqrt{37}}{6}

$\csc (θ) = \frac{\sqrt{37}}{6}$

Step 10

This is the solution to each trig value.

\sin (θ) = \frac{6 \sqrt{37}}{37}

$\sin (θ) = \frac{6 \sqrt{37}}{37}$

\cos (θ) = \frac{\sqrt{37}}{37}

$\cos (θ) = \frac{\sqrt{37}}{37}$

\tan (θ) = 6

$\tan (θ) = 6$

\cot (θ) = \frac{1}{6}

$\cot (θ) = \frac{1}{6}$

\sec (θ) = \sqrt{37}

$\sec (θ) = \sqrt{37}$

\csc (θ) = \frac{\sqrt{37}}{6}

$\csc (θ) = \frac{\sqrt{37}}{6}$