Trigonometry Examples

sec (585 °)

$\sec (585 °)$

Step 1

Remove full rotations of

360

$360$ ° until the angle is between

0

$0$ ° and

360

$360$ °.

sec (225)

$\sec (225)$

Step 2

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

- sec (45)

$- \sec (45)$

Step 3

The exact value of

sec (45)

$\sec (45)$ is

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

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

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

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

Step 4

Multiply

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

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

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

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

- (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})

$- (\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply

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

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

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

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

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

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

Step 5.2

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 5.3

Raise

\sqrt{2}

$\sqrt{2}$ to the power of

1

$1$ .

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

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

Step 5.4

Use the power rule

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

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

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

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

Step 5.5

Add

1

$1$ and

1

$1$ .

- \frac{2 \sqrt{2}}{{\sqrt{2}}^{2}}

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

Step 5.6

Rewrite

{\sqrt{2}}^{2}

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

2

$2$ .

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Step 5.6.1

Use

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

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

\sqrt{2}

$\sqrt{2}$ as

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

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

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

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

Step 5.6.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 5.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

- \frac{2 \sqrt{2}}{2^{\frac{2}{2}}}

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

Step 5.6.4

Cancel the common factor of

2

$2$ .

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Step 5.6.4.1

Cancel the common factor.

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

Step 5.6.4.2

Rewrite the expression.

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

Step 5.6.5

Evaluate the exponent.

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

Step 6

Cancel the common factor of

$2$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Divide

$\sqrt{2}$ by

$1$ .

$- \sqrt{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$- \sqrt{2}$

Decimal Form:

$- 1.41421356 \dots$