Trigonometry Examples

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

Step 1

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

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

Step 2

Cross multiply.

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Step 2.1

Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $\tan (x) \cdot (\sqrt{1 - \sin^{2} (x)})$ .

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Step 2.2.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 2.2.1.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.2.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \sin (x)$ .

$\frac{\sin (x)}{\cos (x)} \cdot \sqrt{(1 + \sin (x)) (1 - \sin (x))} = \sin (x)$

Step 2.2.1.4

Combine $\frac{\sin (x)}{\cos (x)}$ and $\sqrt{(1 + \sin (x)) (1 - \sin (x))}$ .

$\frac{\sin (x) \sqrt{(1 + \sin (x)) (1 - \sin (x))}}{\cos (x)} = \sin (x)$

Step 2.2.1.5

Separate fractions.

$\frac{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}{1} \cdot \frac{\sin (x)}{\cos (x)} = \sin (x)$

Step 2.2.1.6

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}{1} \tan (x) = \sin (x)$

Step 2.2.1.7

Divide $\sqrt{(1 + \sin (x)) (1 - \sin (x))}$ by $1$ .

$\sqrt{(1 + \sin (x)) (1 - \sin (x))} \tan (x) = \sin (x)$

Step 3

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

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

Step 4

Simplify each side of the equation.

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Step 4.1

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

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

Step 4.2

Simplify the left side.

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Step 4.2.1

Simplify ${({((1 + \sin (x)) (1 - \sin (x)))}^{\frac{1}{2}} \tan (x))}^{2}$ .

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Step 4.2.1.1

Expand $(1 + \sin (x)) (1 - \sin (x))$ using the FOIL Method.

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Step 4.2.1.1.1

Apply the distributive property.

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

Step 4.2.1.1.2

Apply the distributive property.

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

Step 4.2.1.1.3

Apply the distributive property.

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

Step 4.2.1.2

Simplify and combine like terms.

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Step 4.2.1.2.1

Simplify each term.

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Step 4.2.1.2.1.1

Multiply $1$ by $1$ .

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

Step 4.2.1.2.1.2

Multiply $- \sin (x)$ by $1$ .

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

Step 4.2.1.2.1.3

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

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

Step 4.2.1.2.1.4

Rewrite using the commutative property of multiplication.

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

Step 4.2.1.2.1.5

Multiply $- \sin (x) \sin (x)$ .

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Step 4.2.1.2.1.5.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.2.1.2.1.5.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 4.2.1.2.1.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2.1.2.1.5.4

Add $1$ and $1$ .

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

Step 4.2.1.2.2

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

${({(1 + 0 - \sin^{2} (x))}^{\frac{1}{2}} \tan (x))}^{2} = \sin^{2} (x)$

Step 4.2.1.2.3

Add $1$ and $0$ .

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

Step 4.2.1.3

Apply pythagorean identity.

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

Step 4.2.1.4

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

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Step 4.2.1.4.1

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

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

Step 4.2.1.4.2

Cancel the common factor of $2$ .

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Step 4.2.1.4.2.1

Cancel the common factor.

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

Step 4.2.1.4.2.2

Rewrite the expression.

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

Step 4.2.1.5

Simplify.

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

Step 4.2.1.6

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 4.2.1.6.1

Reorder $\cos (x)$ and $\tan (x)$ .

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

Step 4.2.1.6.2

Rewrite $\cos (x) \tan (x)$ in terms of sines and cosines.

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

Step 4.2.1.6.3

Cancel the common factors.

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

Step 5

Solve for $x$ .

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Step 5.1

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

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

Step 5.2

Solve for $x$ .

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Step 5.2.1

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 5.2.2

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

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

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

Step 5.2.3

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

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Step 5.2.3.1

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

$x = x$

Step 5.2.3.2

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

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Step 5.2.3.2.1

Subtract $x$ from both sides of the equation.

$x - x = 0$

Step 5.2.3.2.2

Subtract $x$ from $x$ .

$0 = 0$

Step 5.2.3.3

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

All real numbers

Step 5.2.4

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

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Step 5.2.4.1

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

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Step 5.2.4.1.1

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

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

Step 5.2.4.1.2

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

$2 \sin (x) = 0$

Step 5.2.4.2

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

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Step 5.2.4.2.1

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

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

Step 5.2.4.2.2

Simplify the left side.

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Step 5.2.4.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.4.2.2.1.1

Cancel the common factor.

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

Step 5.2.4.2.2.1.2

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

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

Step 5.2.4.2.3

Simplify the right side.

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Step 5.2.4.2.3.1

Divide $0$ by $2$ .

$\sin (x) = 0$

Step 5.2.4.3

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

$x = \arcsin (0)$

Step 5.2.4.4

Simplify the right side.

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Step 5.2.4.4.1

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

$x = 0$

Step 5.2.4.5

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 5.2.4.6

Subtract $0$ from $π$ .

$x = π$

Step 5.2.4.7

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

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Step 5.2.4.7.1

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

$\frac{2 π}{| b |}$

Step 5.2.4.7.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.4.7.3

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

$\frac{2 π}{1}$

Step 5.2.4.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.4.8

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 6

Consolidate the answers.

$x = π n$ , for any integer $n$