Trigonometry Examples

$\sin (\arcsin (\frac{\sqrt{3}}{2}) - \arccos (0))$

Step 1

Simplify each term.

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Step 1.1

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

$\sin (\frac{π}{3} - \arccos (0))$

Step 1.2

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$\sin (\frac{π}{3} - \frac{π}{2})$

Step 2

To write $\frac{π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 3

To write $- \frac{π}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 4

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

Multiply $3$ by $2$ .

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

Step 4.3

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

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

Step 4.4

Multiply $2$ by $3$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Move $2$ to the left of $π$ .

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

Step 6.2

Multiply $3$ by $- 1$ .

$\sin (\frac{2 π - 3 π}{6})$

Step 6.3

Subtract $3 π$ from $2 π$ .

$\sin (\frac{- π}{6})$

Step 7

Move the negative in front of the fraction.

$\sin (- \frac{π}{6})$

Step 8

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

$\sin (\frac{11 π}{6})$

Step 9

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.

$- \sin (\frac{π}{6})$

Step 10

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

$- \frac{1}{2}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$- \frac{1}{2}$

Decimal Form:

$- 0.5$