Trigonometry Examples

sec (θ) = - 2

$\sec (θ) = - 2$

Step 1

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

sec (θ) = \frac{hypotenuse}{adjacent}

$\sec (θ) = \frac{hypotenuse}{adjacent}$

Step 2

Find the opposite side of the unit circle triangle. Since the adjacent side and hypotenuse are known, use the Pythagorean theorem to find the remaining side.

Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}

$Opposite = \sqrt{{hypotenuse}^{2} - {adjacent}^{2}}$

Step 3

Replace the known values in the equation.

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

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

Step 4

Simplify inside the radical.

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Step 4.1

Raise

2

$2$ to the power of

2

$2$ .

Opposite

= \sqrt{4 - {(- 1)}^{2}}

$= \sqrt{4 - {(- 1)}^{2}}$

Step 4.2

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

${(- 1)}^{2}$ by adding the exponents.

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Step 4.2.1

Multiply

- 1

$- 1$ by

{(- 1)}^{2}

${(- 1)}^{2}$ .

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Step 4.2.1.1

Raise

- 1

$- 1$ to the power of

1

$1$ .

Opposite

= \sqrt{4 + (- 1) {(- 1)}^{2}}

$= \sqrt{4 + (- 1) {(- 1)}^{2}}$

Step 4.2.1.2

Use the power rule

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

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

Opposite

= \sqrt{4 + {(- 1)}^{1 + 2}}

$= \sqrt{4 + {(- 1)}^{1 + 2}}$

Opposite

= \sqrt{4 + {(- 1)}^{1 + 2}}

$= \sqrt{4 + {(- 1)}^{1 + 2}}$

Step 4.2.2

Add

1

$1$ and

2

$2$ .

Opposite

= \sqrt{4 + {(- 1)}^{3}}

$= \sqrt{4 + {(- 1)}^{3}}$

Opposite

= \sqrt{4 + {(- 1)}^{3}}

$= \sqrt{4 + {(- 1)}^{3}}$

Step 4.3

Raise

- 1

$- 1$ to the power of

3

$3$ .

Opposite

= \sqrt{4 - 1}

$= \sqrt{4 - 1}$

Step 4.4

Subtract

1

$1$ from

4

$4$ .

Opposite

= \sqrt{3}

$= \sqrt{3}$

Opposite

= \sqrt{3}

$= \sqrt{3}$

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{\sqrt{3}}{2}

$\sin (θ) = \frac{\sqrt{3}}{2}$

sin (θ) = \frac{\sqrt{3}}{2}

$\sin (θ) = \frac{\sqrt{3}}{2}$

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}{2}

$\cos (θ) = \frac{- 1}{2}$

Step 6.3

Move the negative in front of the fraction.

cos (θ) = - \frac{1}{2}

$\cos (θ) = - \frac{1}{2}$

cos (θ) = - \frac{1}{2}

$\cos (θ) = - \frac{1}{2}$

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)

$\tan (θ)$ .

tan (θ) = \frac{opp}{adj}

$\tan (θ) = \frac{opp}{adj}$

Step 7.2

Substitute in the known values.

tan (θ) = \frac{\sqrt{3}}{- 1}

$\tan (θ) = \frac{\sqrt{3}}{- 1}$

Step 7.3

Simplify the value of

tan (θ)

$\tan (θ)$ .

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Step 7.3.1

Move the negative one from the denominator of

\frac{\sqrt{3}}{- 1}

$\frac{\sqrt{3}}{- 1}$ .

tan (θ) = - 1 \cdot \sqrt{3}

$\tan (θ) = - 1 \cdot \sqrt{3}$

Step 7.3.2

Rewrite

- 1 \cdot \sqrt{3}

$- 1 \cdot \sqrt{3}$ as

- \sqrt{3}

$- \sqrt{3}$ .

tan (θ) = - \sqrt{3}

$\tan (θ) = - \sqrt{3}$

tan (θ) = - \sqrt{3}

$\tan (θ) = - \sqrt{3}$

tan (θ) = - \sqrt{3}

$\tan (θ) = - \sqrt{3}$

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)

$\cot (θ)$ .

cot (θ) = \frac{adj}{opp}

$\cot (θ) = \frac{adj}{opp}$

Step 8.2

Substitute in the known values.

cot (θ) = \frac{- 1}{\sqrt{3}}

$\cot (θ) = \frac{- 1}{\sqrt{3}}$

Step 8.3

Simplify the value of

cot (θ)

$\cot (θ)$ .

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Step 8.3.1

Move the negative in front of the fraction.

cot (θ) = - \frac{1}{\sqrt{3}}

$\cot (θ) = - \frac{1}{\sqrt{3}}$

Step 8.3.2

Multiply

\frac{1}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

cot (θ) = - (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})

$\cot (θ) = - (\frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 8.3.3

Combine and simplify the denominator.

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Step 8.3.3.1

Multiply

\frac{1}{\sqrt{3}}

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

\frac{\sqrt{3}}{\sqrt{3}}

$\frac{\sqrt{3}}{\sqrt{3}}$ .

cot (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}

$\cot (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.3.3.2

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

cot (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}

$\cot (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.3.3.3

Raise

\sqrt{3}

$\sqrt{3}$ to the power of

1

$1$ .

cot (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}

$\cot (θ) = - \frac{\sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 8.3.3.4

Use the power rule

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

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

cot (θ) = - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}

$\cot (θ) = - \frac{\sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 8.3.3.5

Add

1

$1$ and

1

$1$ .

cot (θ) = - \frac{\sqrt{3}}{{\sqrt{3}}^{2}}

$\cot (θ) = - \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.3.3.6

Rewrite

{\sqrt{3}}^{2}

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

3

$3$ .

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Step 8.3.3.6.1

Use

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

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

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

cot (θ) = - \frac{\sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}

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

Step 8.3.3.6.2

Apply the power rule and multiply exponents,

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

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

cot (θ) = - \frac{\sqrt{3}}{3^{\frac{1}{2} \cdot 2}}

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

Step 8.3.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

cot (θ) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}

$\cot (θ) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.3.6.4

Cancel the common factor of

2

$2$ .

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Step 8.3.3.6.4.1

Cancel the common factor.

$\cot (θ) = - \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.3.6.4.2

Rewrite the expression.

$\cot (θ) = - \frac{\sqrt{3}}{3}$

Step 8.3.3.6.5

Evaluate the exponent.

$\cot (θ) = - \frac{\sqrt{3}}{3}$

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.

$\csc (θ) = \frac{2}{\sqrt{3}}$

Step 9.3

Simplify the value of

$\csc (θ)$ .

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Step 9.3.1

Multiply

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

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (θ) = \frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 9.3.2

Combine and simplify the denominator.

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Step 9.3.2.1

Multiply

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

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$\csc (θ) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.3.2.2

Raise

$\sqrt{3}$ to the power of

$1$ .

$\csc (θ) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.3.2.3

Raise

$\sqrt{3}$ to the power of

$1$ .

$\csc (θ) = \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 9.3.2.4

Use the power rule

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

$\csc (θ) = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 9.3.2.5

Add

$1$ and

$1$ .

$\csc (θ) = \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 9.3.2.6

Rewrite

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

$3$ .

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Step 9.3.2.6.1

Use

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

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

$\csc (θ) = \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 9.3.2.6.2

Apply the power rule and multiply exponents,

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

$\csc (θ) = \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 9.3.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\csc (θ) = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.3.2.6.4

Cancel the common factor of

$2$ .

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Step 9.3.2.6.4.1

Cancel the common factor.

$\csc (θ) = \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}$

Step 9.3.2.6.4.2

Rewrite the expression.

$\csc (θ) = \frac{2 \sqrt{3}}{3}$

Step 9.3.2.6.5

Evaluate the exponent.

$\csc (θ) = \frac{2 \sqrt{3}}{3}$

Step 10

This is the solution to each trig value.

$\sin (θ) = \frac{\sqrt{3}}{2}$

$\cos (θ) = - \frac{1}{2}$

$\tan (θ) = - \sqrt{3}$

$\cot (θ) = - \frac{\sqrt{3}}{3}$

$\sec (θ) = - 2$

$\csc (θ) = \frac{2 \sqrt{3}}{3}$