Precalculus Examples

tan (θ) = - \frac{2}{7}

$\tan (θ) = - \frac{2}{7}$

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{{(- 2)}^{2} + {(7)}^{2}}

$Hypotenuse = \sqrt{{(- 2)}^{2} + {(7)}^{2}}$

Step 4

Simplify inside the radical.

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Step 4.1

Raise

- 2

$- 2$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{4 + {(7)}^{2}}

$= \sqrt{4 + {(7)}^{2}}$

Step 4.2

Raise

7

$7$ to the power of

2

$2$ .

Hypotenuse

= \sqrt{4 + 49}

$= \sqrt{4 + 49}$

Step 4.3

Add

4

$4$ and

49

$49$ .

Hypotenuse

= \sqrt{53}

$= \sqrt{53}$

Hypotenuse

= \sqrt{53}

$= \sqrt{53}$

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{- 2}{\sqrt{53}}

$\sin (θ) = \frac{- 2}{\sqrt{53}}$

Step 5.3

Simplify the value of

sin (θ)

$\sin (θ)$ .

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Step 5.3.1

Move the negative in front of the fraction.

sin (θ) = - \frac{2}{\sqrt{53}}

$\sin (θ) = - \frac{2}{\sqrt{53}}$

Step 5.3.2

Multiply

\frac{2}{\sqrt{53}}

$\frac{2}{\sqrt{53}}$ by

\frac{\sqrt{53}}{\sqrt{53}}

$\frac{\sqrt{53}}{\sqrt{53}}$ .

sin (θ) = - (\frac{2}{\sqrt{53}} \cdot \frac{\sqrt{53}}{\sqrt{53}})

$\sin (θ) = - (\frac{2}{\sqrt{53}} \cdot \frac{\sqrt{53}}{\sqrt{53}})$

Step 5.3.3

Combine and simplify the denominator.

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Step 5.3.3.1

Multiply

\frac{2}{\sqrt{53}}

$\frac{2}{\sqrt{53}}$ by

\frac{\sqrt{53}}{\sqrt{53}}

$\frac{\sqrt{53}}{\sqrt{53}}$ .

sin (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}

$\sin (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 5.3.3.2

Raise

\sqrt{53}

$\sqrt{53}$ to the power of

1

$1$ .

sin (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}

$\sin (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 5.3.3.3

Raise

\sqrt{53}

$\sqrt{53}$ to the power of

1

$1$ .

sin (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}

$\sin (θ) = - \frac{2 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 5.3.3.4

Use the power rule

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

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

sin (θ) = - \frac{2 \sqrt{53}}{{\sqrt{53}}^{1 + 1}}

$\sin (θ) = - \frac{2 \sqrt{53}}{{\sqrt{53}}^{1 + 1}}$

Step 5.3.3.5

Add

1

$1$ and

1

$1$ .

sin (θ) = - \frac{2 \sqrt{53}}{{\sqrt{53}}^{2}}

$\sin (θ) = - \frac{2 \sqrt{53}}{{\sqrt{53}}^{2}}$

Step 5.3.3.6

Rewrite

{\sqrt{53}}^{2}

${\sqrt{53}}^{2}$ as

53

$53$ .

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Step 5.3.3.6.1

Use

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

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

\sqrt{53}

$\sqrt{53}$ as

53^{\frac{1}{2}}

$53^{\frac{1}{2}}$ .

sin (θ) = - \frac{2 \sqrt{53}}{{(53^{\frac{1}{2}})}^{2}}

$\sin (θ) = - \frac{2 \sqrt{53}}{{(53^{\frac{1}{2}})}^{2}}$

Step 5.3.3.6.2

Apply the power rule and multiply exponents,

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

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

sin (θ) = - \frac{2 \sqrt{53}}{53^{\frac{1}{2} \cdot 2}}

$\sin (θ) = - \frac{2 \sqrt{53}}{53^{\frac{1}{2} \cdot 2}}$

Step 5.3.3.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

sin (θ) = - \frac{2 \sqrt{53}}{53^{\frac{2}{2}}}

$\sin (θ) = - \frac{2 \sqrt{53}}{53^{\frac{2}{2}}}$

Step 5.3.3.6.4

Cancel the common factor of

2

$2$ .

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Step 5.3.3.6.4.1

Cancel the common factor.

$\sin (θ) = - \frac{2 \sqrt{53}}{53^{\frac{2}{2}}}$

Step 5.3.3.6.4.2

Rewrite the expression.

$\sin (θ) = - \frac{2 \sqrt{53}}{53}$

Step 5.3.3.6.5

Evaluate the exponent.

$\sin (θ) = - \frac{2 \sqrt{53}}{53}$

Step 6

Find the value of cosine.

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Step 6.1

Use the definition of cosine to find the value of

$\cos (θ)$ .

$\cos (θ) = \frac{adj}{hyp}$

Step 6.2

Substitute in the known values.

$\cos (θ) = \frac{7}{\sqrt{53}}$

Step 6.3

Simplify the value of

$\cos (θ)$ .

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Step 6.3.1

Multiply

$\frac{7}{\sqrt{53}}$ by

$\frac{\sqrt{53}}{\sqrt{53}}$ .

$\cos (θ) = \frac{7}{\sqrt{53}} \cdot \frac{\sqrt{53}}{\sqrt{53}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply

$\frac{7}{\sqrt{53}}$ by

$\frac{\sqrt{53}}{\sqrt{53}}$ .

$\cos (θ) = \frac{7 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 6.3.2.2

Raise

$\sqrt{53}$ to the power of

$1$ .

$\cos (θ) = \frac{7 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 6.3.2.3

Raise

$\sqrt{53}$ to the power of

$1$ .

$\cos (θ) = \frac{7 \sqrt{53}}{\sqrt{53} \sqrt{53}}$

Step 6.3.2.4

Use the power rule

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

$\cos (θ) = \frac{7 \sqrt{53}}{{\sqrt{53}}^{1 + 1}}$

Step 6.3.2.5

Add

$1$ and

$1$ .

$\cos (θ) = \frac{7 \sqrt{53}}{{\sqrt{53}}^{2}}$

Step 6.3.2.6

Rewrite

${\sqrt{53}}^{2}$ as

$53$ .

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Step 6.3.2.6.1

Use

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

$\sqrt{53}$ as

$53^{\frac{1}{2}}$ .

$\cos (θ) = \frac{7 \sqrt{53}}{{(53^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

Apply the power rule and multiply exponents,

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

$\cos (θ) = \frac{7 \sqrt{53}}{53^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\cos (θ) = \frac{7 \sqrt{53}}{53^{\frac{2}{2}}}$

Step 6.3.2.6.4

Cancel the common factor of

$2$ .

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Step 6.3.2.6.4.1

Cancel the common factor.

$\cos (θ) = \frac{7 \sqrt{53}}{53^{\frac{2}{2}}}$

Step 6.3.2.6.4.2

Rewrite the expression.

$\cos (θ) = \frac{7 \sqrt{53}}{53}$

Step 6.3.2.6.5

Evaluate the exponent.

$\cos (θ) = \frac{7 \sqrt{53}}{53}$

Step 7

Find the value of cotangent.

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Step 7.1

Use the definition of cotangent to find the value of

$\cot (θ)$ .

$\cot (θ) = \frac{adj}{opp}$

Step 7.2

Substitute in the known values.

$\cot (θ) = \frac{7}{- 2}$

Step 7.3

Move the negative in front of the fraction.

$\cot (θ) = - \frac{7}{2}$

Step 8

Find the value of secant.

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Step 8.1

Use the definition of secant to find the value of

$\sec (θ)$ .

$\sec (θ) = \frac{hyp}{adj}$

Step 8.2

Substitute in the known values.

$\sec (θ) = \frac{\sqrt{53}}{7}$

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.

$\csc (θ) = \frac{\sqrt{53}}{- 2}$

Step 9.3

Move the negative in front of the fraction.

$\csc (θ) = - \frac{\sqrt{53}}{2}$

Step 10

This is the solution to each trig value.

$\sin (θ) = - \frac{2 \sqrt{53}}{53}$

$\cos (θ) = \frac{7 \sqrt{53}}{53}$

$\tan (θ) = - \frac{2}{7}$

$\cot (θ) = - \frac{7}{2}$

$\sec (θ) = \frac{\sqrt{53}}{7}$

$\csc (θ) = - \frac{\sqrt{53}}{2}$