Precalculus Examples

\sec (- 300)

$\sec (- 300)$

Step 1

Rewrite

- 300

$- 300$ as an angle where the values of the six trigonometric functions are known divided by

2

$2$ .

\sec (\frac{- 600}{2})

$\sec (\frac{- 600}{2})$

Step 2

Apply the reciprocal identity to

\sec (\frac{- 600}{2})

$\sec (\frac{- 600}{2})$ .

\frac{1}{\cos (\frac{- 600}{2})}

$\frac{1}{\cos (\frac{- 600}{2})}$

Step 3

Apply the cosine half-angle identity

\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}

$\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos (x)}{2}}$ .

\frac{1}{\pm \sqrt{\frac{1 + \cos (- 600)}{2}}}

$\frac{1}{\pm \sqrt{\frac{1 + \cos (- 600)}{2}}}$

Step 4

Change the

\pm

$\pm$ to

+

$+$ because secant is positive in the first quadrant.

\frac{1}{\sqrt{\frac{1 + \cos (- 600)}{2}}}

$\frac{1}{\sqrt{\frac{1 + \cos (- 600)}{2}}}$

Step 5

Simplify

\frac{1}{\sqrt{\frac{1 + \cos (- 600)}{2}}}

$\frac{1}{\sqrt{\frac{1 + \cos (- 600)}{2}}}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Add full rotations of

360

$360$ ° until the angle is between

0

$0$ ° and

360

$360$ °.

\frac{1}{\sqrt{\frac{1 + \cos (120)}{2}}}

$\frac{1}{\sqrt{\frac{1 + \cos (120)}{2}}}$

Step 5.1.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

\frac{1}{\sqrt{\frac{1 - \cos (60)}{2}}}

$\frac{1}{\sqrt{\frac{1 - \cos (60)}{2}}}$

Step 5.1.3

The exact value of

\cos (60)

$\cos (60)$ is

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{\sqrt{\frac{1 - \frac{1}{2}}{2}}}

$\frac{1}{\sqrt{\frac{1 - \frac{1}{2}}{2}}}$

Step 5.1.4

Write

1

$1$ as a fraction with a common denominator.

\frac{1}{\sqrt{\frac{\frac{2}{2} - \frac{1}{2}}{2}}}

$\frac{1}{\sqrt{\frac{\frac{2}{2} - \frac{1}{2}}{2}}}$

Step 5.1.5

Combine the numerators over the common denominator.

\frac{1}{\sqrt{\frac{\frac{2 - 1}{2}}{2}}}

$\frac{1}{\sqrt{\frac{\frac{2 - 1}{2}}{2}}}$

Step 5.1.6

Subtract

1

$1$ from

2

$2$ .

\frac{1}{\sqrt{\frac{\frac{1}{2}}{2}}}

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

\frac{1}{\sqrt{\frac{\frac{1}{2}}{2}}}

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

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply the numerator by the reciprocal of the denominator.

\frac{1}{\sqrt{\frac{1}{2} \cdot \frac{1}{2}}}

$\frac{1}{\sqrt{\frac{1}{2} \cdot \frac{1}{2}}}$

Step 5.2.2

Multiply

\frac{1}{2} \cdot \frac{1}{2}

$\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 5.2.2.1

Multiply

\frac{1}{2}

$\frac{1}{2}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\frac{1}{\sqrt{\frac{1}{2 \cdot 2}}}

$\frac{1}{\sqrt{\frac{1}{2 \cdot 2}}}$

Step 5.2.2.2

Multiply

2

$2$ by

2

$2$ .

\frac{1}{\sqrt{\frac{1}{4}}}

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

\frac{1}{\sqrt{\frac{1}{4}}}

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

Step 5.2.3

Rewrite

\sqrt{\frac{1}{4}}

$\sqrt{\frac{1}{4}}$ as

\frac{\sqrt{1}}{\sqrt{4}}

$\frac{\sqrt{1}}{\sqrt{4}}$ .

\frac{1}{\frac{\sqrt{1}}{\sqrt{4}}}

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

Step 5.2.4

Any root of

1

$1$ is

1

$1$ .

\frac{1}{\frac{1}{\sqrt{4}}}

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

Step 5.2.5

Simplify the denominator.

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Step 5.2.5.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

\frac{1}{\frac{1}{\sqrt{2^{2}}}}

$\frac{1}{\frac{1}{\sqrt{2^{2}}}}$

Step 5.2.5.2

Pull terms out from under the radical, assuming positive real numbers.

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

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

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

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

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

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

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

1 \cdot 2

$1 \cdot 2$

Step 5.4

Multiply

2

$2$ by

1

$1$ .

2

$2$

2

$2$