Trigonometry Examples

$(\frac{3}{5}, - \frac{4}{5})$

Step 1

To find the $\sec (θ)$ between the x-axis and the line between the points $(0, 0)$ and $(\frac{3}{5}, - \frac{4}{5})$ , draw the triangle between the three points $(0, 0)$ , $(\frac{3}{5}, 0)$ , and $(\frac{3}{5}, - \frac{4}{5})$ .

Opposite : $- \frac{4}{5}$

Adjacent : $\frac{3}{5}$

Step 2

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

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Step 2.1

Apply the product rule to $\frac{3}{5}$ .

$\sqrt{\frac{3^{2}}{5^{2}} + {(- \frac{4}{5})}^{2}}$

Step 2.2

Raise $3$ to the power of $2$ .

$\sqrt{\frac{9}{5^{2}} + {(- \frac{4}{5})}^{2}}$

Step 2.3

Raise $5$ to the power of $2$ .

$\sqrt{\frac{9}{25} + {(- \frac{4}{5})}^{2}}$

Step 2.4

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

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Step 2.4.1

Apply the product rule to $- \frac{4}{5}$ .

$\sqrt{\frac{9}{25} + {(- 1)}^{2} {(\frac{4}{5})}^{2}}$

Step 2.4.2

Apply the product rule to $\frac{4}{5}$ .

$\sqrt{\frac{9}{25} + {(- 1)}^{2} \frac{4^{2}}{5^{2}}}$

Step 2.5

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

$\sqrt{\frac{9}{25} + 1 \frac{4^{2}}{5^{2}}}$

Step 2.6

Multiply $\frac{4^{2}}{5^{2}}$ by $1$ .

$\sqrt{\frac{9}{25} + \frac{4^{2}}{5^{2}}}$

Step 2.7

Raise $4$ to the power of $2$ .

$\sqrt{\frac{9}{25} + \frac{16}{5^{2}}}$

Step 2.8

Raise $5$ to the power of $2$ .

$\sqrt{\frac{9}{25} + \frac{16}{25}}$

Step 2.9

Combine the numerators over the common denominator.

$\sqrt{\frac{9 + 16}{25}}$

Step 2.10

Add $9$ and $16$ .

$\sqrt{\frac{25}{25}}$

Step 2.11

Divide $25$ by $25$ .

$\sqrt{1}$

Step 2.12

Any root of $1$ is $1$ .

$1$

Step 3

$\sec (θ) = \frac{Hypotenuse}{Adjacent}$ therefore $\sec (θ) = \frac{1}{\frac{3}{5}}$ .

$\frac{1}{\frac{3}{5}}$

Step 4

Simplify $\sec (θ)$ .

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (θ) = 1 (\frac{5}{3})$

Step 4.2

Multiply $\frac{5}{3}$ by $1$ .

$\sec (θ) = \frac{5}{3}$

Step 5

Approximate the result.

$\sec (θ) = \frac{5}{3} \approx 1.\overline{6}$