Trigonometry Examples

$\sin (\arctan (- 1) + \arccos (\frac{\sqrt{3}}{2}))$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$\sin (- \frac{π}{4} + \arccos (\frac{\sqrt{3}}{2}))$

Step 1.2

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

$\sin (- \frac{π}{4} + \frac{π}{6})$

Step 2

To write $- \frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\sin (- \frac{π}{4} \cdot \frac{3}{3} + \frac{π}{6})$

Step 3

To write $\frac{π}{6}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sin (- \frac{π}{4} \cdot \frac{3}{3} + \frac{π}{6} \cdot \frac{2}{2})$

Step 4

Write each expression with a common denominator of $12$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{π}{4}$ by $\frac{3}{3}$ .

$\sin (- \frac{π \cdot 3}{4 \cdot 3} + \frac{π}{6} \cdot \frac{2}{2})$

Step 4.2

Multiply $4$ by $3$ .

$\sin (- \frac{π \cdot 3}{12} + \frac{π}{6} \cdot \frac{2}{2})$

Step 4.3

Multiply $\frac{π}{6}$ by $\frac{2}{2}$ .

$\sin (- \frac{π \cdot 3}{12} + \frac{π \cdot 2}{6 \cdot 2})$

Step 4.4

Multiply $6$ by $2$ .

$\sin (- \frac{π \cdot 3}{12} + \frac{π \cdot 2}{12})$

Step 5

Combine the numerators over the common denominator.

$\sin (\frac{- π \cdot 3 + π \cdot 2}{12})$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $3$ by $- 1$ .

$\sin (\frac{- 3 π + π \cdot 2}{12})$

Step 6.2

Move $2$ to the left of $π$ .

$\sin (\frac{- 3 π + 2 \cdot π}{12})$

Step 6.3

Add $- 3 π$ and $2 π$ .

$\sin (\frac{- π}{12})$

Step 7

Move the negative in front of the fraction.

$\sin (- \frac{π}{12})$

Step 8

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

$\sin (\frac{23 π}{12})$

Step 9

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

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Step 9.1

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

$- \sin (\frac{π}{12})$

Step 9.2

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

$- \sin (\frac{π}{4} - \frac{π}{6})$

Step 9.3

Apply the difference of angles identity.

$- (\sin (\frac{π}{4}) \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6}))$

Step 9.4

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

$- (\frac{\sqrt{2}}{2} \cos (\frac{π}{6}) - \cos (\frac{π}{4}) \sin (\frac{π}{6}))$

Step 9.5

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (\frac{π}{4}) \sin (\frac{π}{6}))$

Step 9.6

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

$- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (\frac{π}{6}))$

Step 9.7

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

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

Step 9.8

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

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Step 9.8.1

Simplify each term.

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Step 9.8.1.1

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

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Step 9.8.1.1.1

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

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

Step 9.8.1.1.2

Combine using the product rule for radicals.

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

Step 9.8.1.1.3

Multiply $2$ by $3$ .

$- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 9.8.1.1.4

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})$

Step 9.8.1.2

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

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Step 9.8.1.2.1

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

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})$

Step 9.8.1.2.2

Multiply $2$ by $2$ .

$- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})$

Step 9.8.2

Combine the numerators over the common denominator.

$- \frac{\sqrt{6} - \sqrt{2}}{4}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$- \frac{\sqrt{6} - \sqrt{2}}{4}$

Decimal Form:

$- 0.25881904 \dots$