Trigonometry Examples

$\sqrt{1 - \cos^{2} (x)} = \sin (x)$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{1 - \cos^{2} (x)}}^{2} = \sin^{2} (x)$

Step 2

Simplify each side of the equation.

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Step 2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 - \cos^{2} (x)}$ as ${(1 - \cos^{2} (x))}^{\frac{1}{2}}$ .

${({(1 - \cos^{2} (x))}^{\frac{1}{2}})}^{2} = \sin^{2} (x)$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${({(1 - \cos^{2} (x))}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.1

Apply pythagorean identity.

${({(\sin^{2} (x))}^{\frac{1}{2}})}^{2} = \sin^{2} (x)$

Step 2.2.1.2

Apply basic rules of exponents.

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Step 2.2.1.2.1

Multiply the exponents in ${({(\sin^{2} (x))}^{\frac{1}{2}})}^{2}$ .

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Step 2.2.1.2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(\sin^{2} (x))}^{\frac{1}{2} \cdot 2} = \sin^{2} (x)$

Step 2.2.1.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1.2.1

Cancel the common factor.

${(\sin^{2} (x))}^{\frac{1}{2} \cdot 2} = \sin^{2} (x)$

Step 2.2.1.2.1.2.2

Rewrite the expression.

${(\sin^{2} (x))}^{1} = \sin^{2} (x)$

Step 2.2.1.2.2

Multiply the exponents in ${(\sin^{2} (x))}^{1}$ .

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Step 2.2.1.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${\sin (x)}^{2 \cdot 1} = \sin^{2} (x)$

Step 2.2.1.2.2.2

Multiply $2$ by $1$ .

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

Step 3

Solve for $x$ .

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Step 3.1

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

$| \sin (x) | = | \sin (x) |$

Step 3.2

Solve for $x$ .

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Step 3.2.1

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

$\sin (x) = \sin (x)$

$\sin (x) = - \sin (x)$

$- \sin (x) = \sin (x)$

$- \sin (x) = - \sin (x)$

Step 3.2.2

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

$\sin (x) = \sin (x)$

$\sin (x) = - \sin (x)$

Step 3.2.3

Solve $\sin (x) = \sin (x)$ for $x$ .

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Step 3.2.3.1

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

$x = x$

Step 3.2.3.2

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

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Step 3.2.3.2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 3.2.3.2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 3.2.3.3

Since $0 = 0$ , the equation will always be true.

All real numbers

Step 3.2.4

Solve $\sin (x) = - \sin (x)$ for $x$ .

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Step 3.2.4.1

Move all terms containing $\sin (x)$ to the left side of the equation.

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Step 3.2.4.1.1

Add $\sin (x)$ to both sides of the equation.

$\sin (x) + \sin (x) = 0$

Step 3.2.4.1.2

Add $\sin (x)$ and $\sin (x)$ .

$2 \sin (x) = 0$

Step 3.2.4.2

Divide each term in $2 \sin (x) = 0$ by $2$ and simplify.

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Step 3.2.4.2.1

Divide each term in $2 \sin (x) = 0$ by $2$ .

$\frac{2 \sin (x)}{2} = \frac{0}{2}$

Step 3.2.4.2.2

Simplify the left side.

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Step 3.2.4.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.4.2.2.1.1

Cancel the common factor.

$\frac{2 \sin (x)}{2} = \frac{0}{2}$

Step 3.2.4.2.2.1.2

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

$\sin (x) = \frac{0}{2}$

Step 3.2.4.2.3

Simplify the right side.

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Step 3.2.4.2.3.1

Divide $0$ by $2$ .

$\sin (x) = 0$

Step 3.2.4.3

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

$x = \arcsin (0)$

Step 3.2.4.4

Simplify the right side.

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Step 3.2.4.4.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 3.2.4.5

The sine function is positive in the first and second quadrants. To find the second solution, subtract the reference angle from $π$ to find the solution in the second quadrant.

$x = π - 0$

Step 3.2.4.6

Subtract $0$ from $π$ .

$x = π$

Step 3.2.4.7

Find the period of $\sin (x)$ .

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Step 3.2.4.7.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 3.2.4.7.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 3.2.4.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 3.2.4.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2.4.8

The period of the $\sin (x)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2 π n, π + 2 π n$ , for any integer $n$

Step 4

Consolidate the answers.

$x = π n$ , for any integer $n$