Trigonometry Examples

$\sec (- \frac{π}{3})$

Step 1

Replace $\sec (- \frac{π}{3})$ with an equivalent expression $\frac{1}{\cos (- \frac{π}{3})}$ using the fundamental identities.

$\frac{1}{\cos (- \frac{π}{3})}$

Step 2

Use a sum or difference formula on the denominator.

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Step 2.1

The angle $- \frac{π}{3}$ is an angle where the values of the six trigonometric functions are known. Because this is the case, add $0$ to keep the value the same.

$\frac{1}{\cos (- \frac{π}{3} + 0)}$

Step 2.2

Use the difference formula for cosine to simplify the expression. The formula states that $\cos (A - B) = \cos (A) \cos (B) + \sin (A) \sin (B)$ .

$\frac{1}{\cos (0) \cdot \cos (\frac{π}{3}) + \sin (0) \cdot \sin (\frac{π}{3})}$

Step 2.3

Remove parentheses.

$\frac{1}{\cos (0) \cdot \cos (\frac{π}{3}) + \sin (0) \cdot \sin (\frac{π}{3})}$

Step 2.4

Simplify each term.

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Step 2.4.1

The exact value of $\cos (0)$ is $1$ .

$\frac{1}{1 \cdot \cos (\frac{π}{3}) + \sin (0) \cdot \sin (\frac{π}{3})}$

Step 2.4.2

Multiply $\cos (\frac{π}{3})$ by $1$ .

$\frac{1}{\cos (\frac{π}{3}) + \sin (0) \cdot \sin (\frac{π}{3})}$

Step 2.4.3

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$\frac{1}{\frac{1}{2} + \sin (0) \cdot \sin (\frac{π}{3})}$

Step 2.4.4

The exact value of $\sin (0)$ is $0$ .

$\frac{1}{\frac{1}{2} + 0 \cdot \sin (\frac{π}{3})}$

Step 2.4.5

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$\frac{1}{\frac{1}{2} + 0 \cdot \frac{\sqrt{3}}{2}}$

Step 2.4.6

Multiply $0$ by $\frac{\sqrt{3}}{2}$ .

$\frac{1}{\frac{1}{2} + 0}$

Step 2.5

Add $\frac{1}{2}$ and $0$ .

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 2$

Step 3.2

Multiply $2$ by $1$ .

$2$