Trigonometry Examples

$y = 2 \sin (4 x + 45) + 1$

Step 1

Rewrite the equation as $2 \sin (4 x + 45) + 1 = y$ .

$2 \sin (4 x + 45) + 1 = y$

Step 2

Subtract $1$ from both sides of the equation.

$2 \sin (4 x + 45) = y - 1$

Step 3

Divide each term in $2 \sin (4 x + 45) = y - 1$ by $2$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $2 \sin (4 x + 45) = y - 1$ by $2$ .

$\frac{2 \sin (4 x + 45)}{2} = \frac{y}{2} + \frac{- 1}{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{2 \sin (4 x + 45)}{2} = \frac{y}{2} + \frac{- 1}{2}$

Step 3.2.1.2

Divide $\sin (4 x + 45)$ by $1$ .

$\sin (4 x + 45) = \frac{y}{2} + \frac{- 1}{2}$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Move the negative in front of the fraction.

$\sin (4 x + 45) = \frac{y}{2} - \frac{1}{2}$

Step 4

Take the inverse sine of both sides of the equation to extract $x$ from inside the sine.

$4 x + 45 = \arcsin (\frac{y}{2} - \frac{1}{2})$

Step 5

Subtract $45$ from both sides of the equation.

$4 x = \arcsin (\frac{y}{2} - \frac{1}{2}) - 45$

Step 6

Divide each term in $4 x = \arcsin (\frac{y}{2} - \frac{1}{2}) - 45$ by $4$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $4 x = \arcsin (\frac{y}{2} - \frac{1}{2}) - 45$ by $4$ .

$\frac{4 x}{4} = \frac{\arcsin (\frac{y}{2} - \frac{1}{2})}{4} + \frac{- 45}{4}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $4$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{4 x}{4} = \frac{\arcsin (\frac{y}{2} - \frac{1}{2})}{4} + \frac{- 45}{4}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{y}{2} - \frac{1}{2})}{4} + \frac{- 45}{4}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Move the negative in front of the fraction.

$x = \frac{\arcsin (\frac{y}{2} - \frac{1}{2})}{4} - \frac{45}{4}$