Trigonometry Examples

sec (θ) = - \sqrt{10}

$\sec (θ) = - \sqrt{10}$ ,

cot (θ) > 0

$\cot (θ) > 0$

Step 1

The cotangent function is positive in the first and third quadrants. The secant function is negative in the second and third quadrants. The set of solutions for

θ

$θ$ are limited to the third quadrant since that is the only quadrant found in both sets.

Solution is in the third quadrant.

Step 2

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 3

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 4

Replace the known values in the equation.

Opposite = - \sqrt{{(\sqrt{10})}^{2} - {(- 1)}^{2}}

$Opposite = - \sqrt{{(\sqrt{10})}^{2} - {(- 1)}^{2}}$

Step 5

Simplify inside the radical.

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Step 5.1

Negate

\sqrt{{(\sqrt{10})}^{2} - {(- 1)}^{2}}

$\sqrt{{(\sqrt{10})}^{2} - {(- 1)}^{2}}$ .

Opposite

= - \sqrt{{(\sqrt{10})}^{2} - {(- 1)}^{2}}

$= - \sqrt{{(\sqrt{10})}^{2} - {(- 1)}^{2}}$

Step 5.2

Rewrite

{\sqrt{10}}^{2}

${\sqrt{10}}^{2}$ as

10

$10$ .

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Step 5.2.1

Use

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

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

\sqrt{10}

$\sqrt{10}$ as

10^{\frac{1}{2}}

$10^{\frac{1}{2}}$ .

Opposite

= - \sqrt{{(10^{\frac{1}{2}})}^{2} - {(- 1)}^{2}}

$= - \sqrt{{(10^{\frac{1}{2}})}^{2} - {(- 1)}^{2}}$

Step 5.2.2

Apply the power rule and multiply exponents,

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

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

Opposite

= - \sqrt{10^{\frac{1}{2} \cdot 2} - {(- 1)}^{2}}

$= - \sqrt{10^{\frac{1}{2} \cdot 2} - {(- 1)}^{2}}$

Step 5.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

Opposite

= - \sqrt{10^{\frac{2}{2}} - {(- 1)}^{2}}

$= - \sqrt{10^{\frac{2}{2}} - {(- 1)}^{2}}$

Step 5.2.4

Cancel the common factor of

2

$2$ .

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Step 5.2.4.1

Cancel the common factor.

Opposite

$= - \sqrt{10^{\frac{2}{2}} - {(- 1)}^{2}}$

Step 5.2.4.2

Rewrite the expression.

Opposite

$= - \sqrt{10 - {(- 1)}^{2}}$

Opposite

$= - \sqrt{10 - {(- 1)}^{2}}$

Step 5.2.5

Evaluate the exponent.

Opposite

$= - \sqrt{10 - {(- 1)}^{2}}$

Opposite

$= - \sqrt{10 - {(- 1)}^{2}}$

Step 5.3

Multiply

$- 1$ by

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

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Step 5.3.1

Multiply

$- 1$ by

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

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Step 5.3.1.1

Raise

$- 1$ to the power of

$1$ .

Opposite

$= - \sqrt{10 + (- 1) {(- 1)}^{2}}$

Step 5.3.1.2

Use the power rule

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

Opposite

$= - \sqrt{10 + {(- 1)}^{1 + 2}}$

Opposite

$= - \sqrt{10 + {(- 1)}^{1 + 2}}$

Step 5.3.2

Add

$1$ and

$2$ .

Opposite

$= - \sqrt{10 + {(- 1)}^{3}}$

Opposite

$= - \sqrt{10 + {(- 1)}^{3}}$

Step 5.4

Raise

$- 1$ to the power of

$3$ .

Opposite

$= - \sqrt{10 - 1}$

Step 5.5

Subtract

$1$ from

$10$ .

Opposite

$= - \sqrt{9}$

Step 5.6

Rewrite

$9$ as

$3^{2}$ .

Opposite

$= - \sqrt{3^{2}}$

Step 5.7

Pull terms out from under the radical, assuming positive real numbers.

Opposite

$= - 1 \cdot 3$

Step 5.8

Multiply

$- 1$ by

$3$ .

Opposite

$= - 3$

Opposite

$= - 3$

Step 6

Find the value of sine.

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Step 6.1

Use the definition of sine to find the value of

$\sin (θ)$ .

$\sin (θ) = \frac{opp}{hyp}$

Step 6.2

Substitute in the known values.

$\sin (θ) = \frac{- 3}{\sqrt{10}}$

Step 6.3

Simplify the value of

$\sin (θ)$ .

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Step 6.3.1

Move the negative in front of the fraction.

$\sin (θ) = - \frac{3}{\sqrt{10}}$

Step 6.3.2

Multiply

$\frac{3}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (θ) = - (\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 6.3.3

Combine and simplify the denominator.

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Step 6.3.3.1

Multiply

$\frac{3}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 6.3.3.2

Raise

$\sqrt{10}$ to the power of

$1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 6.3.3.3

Raise

$\sqrt{10}$ to the power of

$1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 6.3.3.4

Use the power rule

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

$\sin (θ) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 6.3.3.5

Add

$1$ and

$1$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 6.3.3.6

Rewrite

${\sqrt{10}}^{2}$ as

$10$ .

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Step 6.3.3.6.1

Use

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

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 6.3.3.6.2

Apply the power rule and multiply exponents,

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

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 6.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

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.

$\sin (θ) = - \frac{3 \sqrt{10}}{10^{\frac{2}{2}}}$

Step 6.3.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

Step 6.3.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

Step 7

Find the value of cosine.

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Step 7.1

Use the definition of cosine to find the value of

$\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Step 7.2

Substitute in the known values.

$\cos (θ) = \frac{- 1}{\sqrt{10}}$

Step 7.3

Simplify the value of

$\cos (θ)$ .

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Step 7.3.1

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{\sqrt{10}}$

Step 7.3.2

Multiply

$\frac{1}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = - (\frac{1}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

Multiply

$\frac{1}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 7.3.3.2

Raise

$\sqrt{10}$ to the power of

$1$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 7.3.3.3

Raise

$\sqrt{10}$ to the power of

$1$ .

$\cos (θ) = - \frac{\sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 7.3.3.4

Use the power rule

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

$\cos (θ) = - \frac{\sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 7.3.3.5

Add

$1$ and

$1$ .

$\cos (θ) = - \frac{\sqrt{10}}{{\sqrt{10}}^{2}}$

Step 7.3.3.6

Rewrite

${\sqrt{10}}^{2}$ as

$10$ .

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Step 7.3.3.6.1

Use

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

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

$\cos (θ) = - \frac{\sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

Apply the power rule and multiply exponents,

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

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

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.

$\cos (θ) = - \frac{\sqrt{10}}{10^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\cos (θ) = - \frac{\sqrt{10}}{10}$

Step 7.3.3.6.5

Evaluate the exponent.

$\cos (θ) = - \frac{\sqrt{10}}{10}$

Step 8

Find the value of tangent.

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Step 8.1

Use the definition of tangent to find the value of

$\tan (θ)$ .

$\tan (θ) = \frac{opp}{adj}$

Step 8.2

Substitute in the known values.

$\tan (θ) = \frac{- 3}{- 1}$

Step 8.3

Divide

$- 3$ by

$- 1$ .

$\tan (θ) = 3$

Step 9

Find the value of cotangent.

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Step 9.1

Use the definition of cotangent to find the value of

$\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Step 9.2

Substitute in the known values.

$\cot (θ) = \frac{- 1}{- 3}$

Step 9.3

Dividing two negative values results in a positive value.

$\cot (θ) = \frac{1}{3}$

Step 10

Find the value of cosecant.

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Step 10.1

Use the definition of cosecant to find the value of

$\csc (θ)$ .

$\csc (θ) = \frac{hyp}{opp}$

Step 10.2

Substitute in the known values.

$\csc (θ) = \frac{\sqrt{10}}{- 3}$

Step 10.3

Move the negative in front of the fraction.

$\csc (θ) = - \frac{\sqrt{10}}{3}$

Step 11

This is the solution to each trig value.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

$\cos (θ) = - \frac{\sqrt{10}}{10}$

$\tan (θ) = 3$

$\cot (θ) = \frac{1}{3}$

$\sec (θ) = - \sqrt{10}$

$\csc (θ) = - \frac{\sqrt{10}}{3}$