Trigonometry Examples

$f (x) = - 0.5 \cos (- \frac{π}{12}) + 0.27$

Step 1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$f (x) = - 0.5 \cos (\frac{23 π}{12}) + 0.27$

Step 2

The exact value of $\cos (\frac{23 π}{12})$ is $\frac{\sqrt{6} + \sqrt{2}}{4}$ .

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Step 2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$f (x) = - 0.5 \cos (\frac{π}{12}) + 0.27$

Step 2.2

Split $\frac{π}{12}$ into two angles where the values of the six trigonometric functions are known.

$f (x) = - 0.5 \cos (\frac{π}{4} - \frac{π}{6}) + 0.27$

Step 2.3

Apply the difference of angles identity $\cos (x - y) = \cos (x) \cos (y) + \sin (x) \sin (y)$ .

$f (x) = - 0.5 (\cos (\frac{π}{4}) \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) + 0.27$

Step 2.4

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (x) = - 0.5 (\frac{\sqrt{2}}{2} \cdot \cos (\frac{π}{6}) + \sin (\frac{π}{4}) \sin (\frac{π}{6})) + 0.27$

Step 2.5

The exact value of $\cos (\frac{π}{6})$ is $\frac{\sqrt{3}}{2}$ .

$f (x) = - 0.5 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (\frac{π}{4}) \sin (\frac{π}{6})) + 0.27$

Step 2.6

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$f (x) = - 0.5 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \sin (\frac{π}{6})) + 0.27$

Step 2.7

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$f (x) = - 0.5 (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + 0.27$

Step 2.8

Simplify $\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}$ .

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Step 2.8.1

Simplify each term.

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Step 2.8.1.1

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

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Step 2.8.1.1.1

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

$f (x) = - 0.5 (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + 0.27$

Step 2.8.1.1.2

Combine using the product rule for radicals.

$f (x) = - 0.5 (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + 0.27$

Step 2.8.1.1.3

Multiply $2$ by $3$ .

$f (x) = - 0.5 (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + 0.27$

Step 2.8.1.1.4

Multiply $2$ by $2$ .

$f (x) = - 0.5 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + 0.27$

Step 2.8.1.2

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

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Step 2.8.1.2.1

Multiply $\frac{\sqrt{2}}{2}$ by $\frac{1}{2}$ .

$f (x) = - 0.5 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) + 0.27$

Step 2.8.1.2.2

Multiply $2$ by $2$ .

$f (x) = - 0.5 (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) + 0.27$

Step 2.8.2

Combine the numerators over the common denominator.

$f (x) = - 0.5 \frac{\sqrt{6} + \sqrt{2}}{4} + 0.27$

Step 3

Cancel the common factor of $0.5$ .

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Step 3.1

Factor $0.5$ out of $- 0.5$ .

$f (x) = 0.5 (- 1) (\frac{\sqrt{6} + \sqrt{2}}{4}) + 0.27$

Step 3.2

Factor $0.5$ out of $4$ .

$f (x) = 0.5 \cdot (- 1 \frac{\sqrt{6} + \sqrt{2}}{0.5 \cdot 8}) + 0.27$

Step 3.3

Cancel the common factor.

$f (x) = 0.5 \cdot (- 1 \frac{\sqrt{6} + \sqrt{2}}{0.5 \cdot 8}) + 0.27$

Step 3.4

Rewrite the expression.

$f (x) = - 1 \frac{\sqrt{6} + \sqrt{2}}{8} + 0.27$

Step 4

Rewrite $- 1 \frac{\sqrt{6} + \sqrt{2}}{8}$ as $- \frac{\sqrt{6} + \sqrt{2}}{8}$ .

$f (x) = - \frac{\sqrt{6} + \sqrt{2}}{8} + 0.27$

Step 5

To write $0.27$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$f (x) = - \frac{\sqrt{6} + \sqrt{2}}{8} + 0.27 \cdot \frac{8}{8}$

Step 6

Combine $0.27$ and $\frac{8}{8}$ .

$f (x) = - \frac{\sqrt{6} + \sqrt{2}}{8} + \frac{0.27 \cdot 8}{8}$

Step 7

Combine the numerators over the common denominator.

$f (x) = \frac{- (\sqrt{6} + \sqrt{2}) + 0.27 \cdot 8}{8}$

Step 8

Rewrite $\frac{- (\sqrt{6} + \sqrt{2}) + 0.27 \cdot 8}{8}$ in a factored form.

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Step 8.1

Apply the distributive property.

$f (x) = \frac{- \sqrt{6} - \sqrt{2} + 0.27 \cdot 8}{8}$

Step 8.2

Multiply $0.27$ by $8$ .

$f (x) = \frac{- \sqrt{6} - \sqrt{2} + 2.16}{8}$

Step 8.3

Add $- \sqrt{6}$ and $2.16$ .

$f (x) = \frac{- 0.28948974 - \sqrt{2}}{8}$

Step 8.4

Subtract $\sqrt{2}$ from $- 0.28948974$ .

$f (x) = \frac{- 1.7037033}{8}$

Step 8.5

Divide $- 1.7037033$ by $8$ .

$f (x) = - 0.21296291$