Trigonometry Examples

$\sin (\frac{7 π}{6} - \frac{31 π}{3})$

Step 1

To write $- \frac{31 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\sin (\frac{7 π}{6} - \frac{31 π}{3} \cdot \frac{2}{2})$

Step 2

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

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Step 2.1

Multiply $\frac{31 π}{3}$ by $\frac{2}{2}$ .

$\sin (\frac{7 π}{6} - \frac{31 π \cdot 2}{3 \cdot 2})$

Step 2.2

Multiply $3$ by $2$ .

$\sin (\frac{7 π}{6} - \frac{31 π \cdot 2}{6})$

Step 3

Combine the numerators over the common denominator.

$\sin (\frac{7 π - 31 π \cdot 2}{6})$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $2$ by $- 31$ .

$\sin (\frac{7 π - 62 π}{6})$

Step 4.2

Subtract $62 π$ from $7 π$ .

$\sin (\frac{- 55 π}{6})$

Step 5

Move the negative in front of the fraction.

$\sin (- \frac{55 π}{6})$

Step 6

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

$\sin (\frac{5 π}{6})$

Step 7

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

$\sin (\frac{π}{6})$

Step 8

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

$\frac{1}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$