Trigonometry Examples

$(- \frac{20}{29}, \frac{21}{29})$

Step 1

To find the $\sec (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(- \frac{20}{29}, \frac{21}{29})$ , draw the triangle between the three points $(0, 0)$ , $(- \frac{20}{29}, 0)$ , and $(- \frac{20}{29}, \frac{21}{29})$ .

Opposite : $\frac{21}{29}$

Adjacent : $- \frac{20}{29}$

Step 2

Find the hypotenuse using Pythagorean theorem $c = \sqrt{a^{2} + b^{2}}$ .

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Step 2.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 2.1.1

Apply the product rule to $- \frac{20}{29}$ .

$\sqrt{{(- 1)}^{2} {(\frac{20}{29})}^{2} + {(\frac{21}{29})}^{2}}$

Step 2.1.2

Apply the product rule to $\frac{20}{29}$ .

$\sqrt{{(- 1)}^{2} \frac{20^{2}}{29^{2}} + {(\frac{21}{29})}^{2}}$

Step 2.2

Raise $- 1$ to the power of $2$ .

$\sqrt{1 \frac{20^{2}}{29^{2}} + {(\frac{21}{29})}^{2}}$

Step 2.3

Multiply $\frac{20^{2}}{29^{2}}$ by $1$ .

$\sqrt{\frac{20^{2}}{29^{2}} + {(\frac{21}{29})}^{2}}$

Step 2.4

Raise $20$ to the power of $2$ .

$\sqrt{\frac{400}{29^{2}} + {(\frac{21}{29})}^{2}}$

Step 2.5

Raise $29$ to the power of $2$ .

$\sqrt{\frac{400}{841} + {(\frac{21}{29})}^{2}}$

Step 2.6

Apply the product rule to $\frac{21}{29}$ .

$\sqrt{\frac{400}{841} + \frac{21^{2}}{29^{2}}}$

Step 2.7

Raise $21$ to the power of $2$ .

$\sqrt{\frac{400}{841} + \frac{441}{29^{2}}}$

Step 2.8

Raise $29$ to the power of $2$ .

$\sqrt{\frac{400}{841} + \frac{441}{841}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{400 + 441}{841}}$

Step 2.10

Add $400$ and $441$ .

$\sqrt{\frac{841}{841}}$

Step 2.11

Divide $841$ by $841$ .

$\sqrt{1}$

Step 2.12

Any root of $1$ is $1$ .

$1$

Step 3

$\sec (θ) = \frac{Hypotenuse}{Adjacent}$ therefore $\sec (θ) = \frac{1}{- \frac{20}{29}}$ .

$\frac{1}{- \frac{20}{29}}$

Step 4

Simplify $\sec (θ)$ .

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Step 4.1

Cancel the common factor of $1$ and $- 1$ .

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Step 4.1.1

Rewrite $1$ as $- 1 (- 1)$ .

$\sec (θ) = \frac{- 1 \cdot - 1}{- \frac{20}{29}}$

Step 4.1.2

Move the negative in front of the fraction.

$\sec (θ) = - \frac{1}{\frac{20}{29}}$

Step 4.2

Multiply the numerator by the reciprocal of the denominator.

$\sec (θ) = - (1 (\frac{29}{20}))$

Step 4.3

Multiply $\frac{29}{20}$ by $1$ .

$\sec (θ) = - \frac{29}{20}$

Step 5

Approximate the result.

$\sec (θ) = - \frac{29}{20} \approx - 1.45$