Trigonometry Examples

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

Step 1

Since the exponents are equal, the bases of the exponents on both sides of the equation must be equal.

$| \sin (\frac{x}{2}) | = | \sin (x) |$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the absolute value equation as four equations without absolute value bars.

$\sin (\frac{x}{2}) = \sin (x)$

$\sin (\frac{x}{2}) = - \sin (x)$

$- \sin (\frac{x}{2}) = \sin (x)$

$- \sin (\frac{x}{2}) = - \sin (x)$

Step 2.2

After simplifying, there are only two unique equations to be solved.

$\sin (\frac{x}{2}) = \sin (x)$

$\sin (\frac{x}{2}) = - \sin (x)$

Step 2.3

Solve $\sin (\frac{x}{2}) = \sin (x)$ for $x$ .

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Step 2.3.1

For the two functions to be equal, the arguments of each must be equal.

$\frac{x}{2} = x$

Step 2.3.2

Multiply both sides by $2$ .

$\frac{x}{2} \cdot 2 = x \cdot 2$

Step 2.3.3

Simplify.

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Step 2.3.3.1

Simplify the left side.

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Step 2.3.3.1.1

Cancel the common factor of $2$ .

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Step 2.3.3.1.1.1

Cancel the common factor.

$\frac{x}{2} \cdot 2 = x \cdot 2$

Step 2.3.3.1.1.2

Rewrite the expression.

$x = x \cdot 2$

Step 2.3.3.2

Simplify the right side.

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Step 2.3.3.2.1

Move $2$ to the left of $x$ .

$x = 2 x$

Step 2.3.4

Solve for $x$ .

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Step 2.3.4.1

Move all terms containing $x$ to the left side of the equation.

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Step 2.3.4.1.1

Subtract $2 x$ from both sides of the equation.

$x - 2 x = 0$

Step 2.3.4.1.2

Subtract $2 x$ from $x$ .

$- x = 0$

Step 2.3.4.2

Divide each term in $- x = 0$ by $- 1$ and simplify.

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Step 2.3.4.2.1

Divide each term in $- x = 0$ by $- 1$ .

$\frac{- x}{- 1} = \frac{0}{- 1}$

Step 2.3.4.2.2

Simplify the left side.

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Step 2.3.4.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{0}{- 1}$

Step 2.3.4.2.2.2

Divide $x$ by $1$ .

$x = \frac{0}{- 1}$

Step 2.3.4.2.3

Simplify the right side.

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Step 2.3.4.2.3.1

Divide $0$ by $- 1$ .

$x = 0$

Step 2.4

List all of the solutions.

$x = 0, 0$