Trigonometry Examples

(\frac{1}{2}, - \frac{\sqrt{3}}{2})

$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$

Step 1

To find the

sec (θ)

$\sec (θ)$ between the x-axis and the line between the points

(0, 0)

$(0, 0)$ and

(\frac{1}{2}, - \frac{\sqrt{3}}{2})

$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ , draw the triangle between the three points

(0, 0)

$(0, 0)$ ,

(\frac{1}{2}, 0)

$(\frac{1}{2}, 0)$ , and

(\frac{1}{2}, - \frac{\sqrt{3}}{2})

$(\frac{1}{2}, - \frac{\sqrt{3}}{2})$ .

Opposite :

- \frac{\sqrt{3}}{2}

$- \frac{\sqrt{3}}{2}$

Adjacent :

\frac{1}{2}

$\frac{1}{2}$

Step 2

Find the hypotenuse using Pythagorean theorem

c = \sqrt{a^{2} + b^{2}}

$c = \sqrt{a^{2} + b^{2}}$ .

Tap for more steps...

Step 2.1

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

\sqrt{\frac{1^{2}}{2^{2}} + {(- \frac{\sqrt{3}}{2})}^{2}}

$\sqrt{\frac{1^{2}}{2^{2}} + {(- \frac{\sqrt{3}}{2})}^{2}}$

Step 2.2

One to any power is one.

\sqrt{\frac{1}{2^{2}} + {(- \frac{\sqrt{3}}{2})}^{2}}

$\sqrt{\frac{1}{2^{2}} + {(- \frac{\sqrt{3}}{2})}^{2}}$

Step 2.3

Raise

2

$2$ to the power of

2

$2$ .

\sqrt{\frac{1}{4} + {(- \frac{\sqrt{3}}{2})}^{2}}

$\sqrt{\frac{1}{4} + {(- \frac{\sqrt{3}}{2})}^{2}}$

Step 2.4

Use the power rule

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

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

Tap for more steps...

Step 2.4.1

Apply the product rule to

- \frac{\sqrt{3}}{2}

$- \frac{\sqrt{3}}{2}$ .

\sqrt{\frac{1}{4} + {(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2}}

$\sqrt{\frac{1}{4} + {(- 1)}^{2} {(\frac{\sqrt{3}}{2})}^{2}}$

Step 2.4.2

Apply the product rule to

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

\sqrt{\frac{1}{4} + {(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}}}

$\sqrt{\frac{1}{4} + {(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}}}$

\sqrt{\frac{1}{4} + {(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}}}

$\sqrt{\frac{1}{4} + {(- 1)}^{2} \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2.5

Simplify the expression.

Tap for more steps...

Step 2.5.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

\sqrt{\frac{1}{4} + 1 \frac{{\sqrt{3}}^{2}}{2^{2}}}

$\sqrt{\frac{1}{4} + 1 \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2.5.2

Multiply

\frac{{\sqrt{3}}^{2}}{2^{2}}

$\frac{{\sqrt{3}}^{2}}{2^{2}}$ by

1

$1$ .

\sqrt{\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}}

$\sqrt{\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}}$

\sqrt{\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}}

$\sqrt{\frac{1}{4} + \frac{{\sqrt{3}}^{2}}{2^{2}}}$

Step 2.6

Rewrite

{\sqrt{3}}^{2}

${\sqrt{3}}^{2}$ as

3

$3$ .

Tap for more steps...

Step 2.6.1

Use

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

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

\sqrt{3}

$\sqrt{3}$ as

3^{\frac{1}{2}}

$3^{\frac{1}{2}}$ .

\sqrt{\frac{1}{4} + \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}

$\sqrt{\frac{1}{4} + \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 2.6.2

Apply the power rule and multiply exponents,

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

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

\sqrt{\frac{1}{4} + \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}

$\sqrt{\frac{1}{4} + \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 2.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

\sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}

$\sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.4

Cancel the common factor of

2

$2$ .

Tap for more steps...

Step 2.6.4.1

Cancel the common factor.

$\sqrt{\frac{1}{4} + \frac{3^{\frac{2}{2}}}{2^{2}}}$

Step 2.6.4.2

Rewrite the expression.

$\sqrt{\frac{1}{4} + \frac{3^{1}}{2^{2}}}$

Step 2.6.5

Evaluate the exponent.

$\sqrt{\frac{1}{4} + \frac{3}{2^{2}}}$

Step 2.7

Simplify the expression.

Tap for more steps...

Step 2.7.1

Raise

$2$ to the power of

$2$ .

$\sqrt{\frac{1}{4} + \frac{3}{4}}$

Step 2.7.2

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 3}{4}}$

Step 2.7.3

Add

$1$ and

$3$ .

$\sqrt{\frac{4}{4}}$

Step 2.7.4

Divide

$4$ by

$4$ .

$\sqrt{1}$

Step 2.7.5

Any root of

$1$ is

$1$ .

$1$

Step 3

$\sec (θ) = \frac{Hypotenuse}{Adjacent}$ therefore

$\sec (θ) = \frac{1}{\frac{1}{2}}$ .

$\frac{1}{\frac{1}{2}}$

Step 4

Simplify

$\sec (θ)$ .

Tap for more steps...

Step 4.1

Multiply the numerator by the reciprocal of the denominator.

$\sec (θ) = 1 \cdot 2$

Step 4.2

Multiply

$2$ by

$1$ .

$\sec (θ) = 2$