Trigonometry Examples

$r \cos (x) r \sin (x) = 2$

Step 1

Multiply each term in $r \cos (x) r \sin (x) = 2$ by $2$ to eliminate the fractions.

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Step 1.1

Multiply each term in $r \cos (x) r \sin (x) = 2$ by $2$ .

$r \cos (x) r \sin (x) \cdot 2 = 2 \cdot 2$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Add parentheses.

$r \cos (x) r (\sin (x) \cdot 2) = 2 \cdot 2$

Step 1.2.1.2

Add parentheses.

$r \cos (x) (r (\sin (x) \cdot 2)) = 2 \cdot 2$

Step 1.2.1.3

Add parentheses.

$r (\cos (x) (r (\sin (x) \cdot 2))) = 2 \cdot 2$

Step 1.2.1.4

Reorder $\cos (x)$ and $r (\sin (x) \cdot 2)$ .

$r (r (\sin (x) \cdot 2) \cos (x)) = 2 \cdot 2$

Step 1.2.1.5

Move parentheses.

$r (r (\sin (x) \cdot 2 \cos (x))) = 2 \cdot 2$

Step 1.2.1.6

Reorder $\sin (x)$ and $2$ .

$r (r (2 \cdot \sin (x) \cos (x))) = 2 \cdot 2$

Step 1.2.2

Apply the sine double-angle identity.

$r (r \sin (2 x)) = 2 \cdot 2$

Step 1.2.3

Multiply $r$ by $r$ by adding the exponents.

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Step 1.2.3.1

Move $r$ .

$r \cdot r \sin (2 x) = 2 \cdot 2$

Step 1.2.3.2

Multiply $r$ by $r$ .

$r^{2} \sin (2 x) = 2 \cdot 2$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply $2$ by $2$ .

$r^{2} \sin (2 x) = 4$

Step 2

Divide each term in $r^{2} \sin (2 x) = 4$ by $r^{2}$ and simplify.

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Step 2.1

Divide each term in $r^{2} \sin (2 x) = 4$ by $r^{2}$ .

$\frac{r^{2} \sin (2 x)}{r^{2}} = \frac{4}{r^{2}}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $r^{2}$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{r^{2} \sin (2 x)}{r^{2}} = \frac{4}{r^{2}}$

Step 2.2.1.2

Divide $\sin (2 x)$ by $1$ .

$\sin (2 x) = \frac{4}{r^{2}}$

Step 3

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

$2 x = \arcsin (\frac{4}{r^{2}})$

Step 4

Divide each term in $2 x = \arcsin (\frac{4}{r^{2}})$ by $2$ and simplify.

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Step 4.1

Divide each term in $2 x = \arcsin (\frac{4}{r^{2}})$ by $2$ .

$\frac{2 x}{2} = \frac{\arcsin (\frac{4}{r^{2}})}{2}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $2$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{\arcsin (\frac{4}{r^{2}})}{2}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\arcsin (\frac{4}{r^{2}})}{2}$