Trigonometry Examples

$\sec (θ) = - \sqrt{10}$ , $\cot (θ) > 0$

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 $θ$ 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.

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

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.

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

Step 4

Replace the known values in the equation.

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

Step 5

Simplify inside the radical.

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Step 5.1

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

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

Step 5.2

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

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Step 5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

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

Step 5.2.2

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

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

Step 5.2.3

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

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

Step 5.2.4

Cancel the common factor of $2$ .

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

Step 5.2.5

Evaluate the exponent.

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

Step 5.3.2

Add $1$ and $2$ .

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 $= - 3$

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.

$\sin (θ) = - \frac{3 \sqrt{10}}{10}$

$\cos (θ) = - \frac{\sqrt{10}}{10}$

$\tan (θ) = 3$

$\cot (θ) = \frac{1}{3}$

$\sec (θ) = - \sqrt{10}$

$\csc (θ) = - \frac{\sqrt{10}}{3}$