Trigonometry Examples

$\sqrt{2} \sin (x) \cos (x) - \sin (x) - \sqrt{2} \cos (x) + 1 = 0$

Step 1

Factor $\sqrt{2} \sin (x) \cos (x) - \sin (x) - \sqrt{2} \cos (x) + 1$ .

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Step 1.1

Factor out the greatest common factor from each group.

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Step 1.1.1

Group the first two terms and the last two terms.

$\sqrt{2} \sin (x) \cos (x) - \sin (x) - \sqrt{2} \cos (x) + 1 = 0$

Step 1.1.2

Factor out the greatest common factor (GCF) from each group.

$\sin (x) (\sqrt{2} \cos (x) - 1) - (\sqrt{2} \cos (x) - 1) = 0$

Step 1.2

Factor the polynomial by factoring out the greatest common factor, $\sqrt{2} \cos (x) - 1$ .

$(\sqrt{2} \cos (x) - 1) (\sin (x) - 1) = 0$

Step 2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$\sqrt{2} \cos (x) - 1 = 0$

$\sin (x) - 1 = 0$

Step 3

Set $\sqrt{2} \cos (x) - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\sqrt{2} \cos (x) - 1$ equal to $0$ .

$\sqrt{2} \cos (x) - 1 = 0$

Step 3.2

Solve $\sqrt{2} \cos (x) - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

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

Step 3.2.2

Divide each term in $\sqrt{2} \cos (x) = 1$ by $\sqrt{2}$ and simplify.

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Step 3.2.2.1

Divide each term in $\sqrt{2} \cos (x) = 1$ by $\sqrt{2}$ .

$\frac{\sqrt{2} \cos (x)}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

Step 3.2.2.2

Simplify the left side.

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Step 3.2.2.2.1

Cancel the common factor of $\sqrt{2}$ .

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Step 3.2.2.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} \cos (x)}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

Step 3.2.2.2.1.2

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

$\cos (x) = \frac{1}{\sqrt{2}}$

Step 3.2.2.3

Simplify the right side.

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Step 3.2.2.3.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 3.2.2.3.2

Combine and simplify the denominator.

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Step 3.2.2.3.2.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\cos (x) = \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 3.2.2.3.2.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.2.2.3.2.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 3.2.2.3.2.4

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

$\cos (x) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 3.2.2.3.2.5

Add $1$ and $1$ .

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

Step 3.2.2.3.2.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 3.2.2.3.2.6.1

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

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

Step 3.2.2.3.2.6.2

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

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

Step 3.2.2.3.2.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 3.2.2.3.2.6.4

Cancel the common factor of $2$ .

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Step 3.2.2.3.2.6.4.1

Cancel the common factor.

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

Step 3.2.2.3.2.6.4.2

Rewrite the expression.

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

Step 3.2.2.3.2.6.5

Evaluate the exponent.

$\cos (x) = \frac{\sqrt{2}}{2}$

Step 3.2.3

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

$x = \arccos (\frac{\sqrt{2}}{2})$

Step 3.2.4

Simplify the right side.

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Step 3.2.4.1

The exact value of $\arccos (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x = \frac{π}{4}$

Step 3.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$x = 2 π - \frac{π}{4}$

Step 3.2.6

Simplify $2 π - \frac{π}{4}$ .

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Step 3.2.6.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = 2 π \cdot \frac{4}{4} - \frac{π}{4}$

Step 3.2.6.2

Combine fractions.

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Step 3.2.6.2.1

Combine $2 π$ and $\frac{4}{4}$ .

$x = \frac{2 π \cdot 4}{4} - \frac{π}{4}$

Step 3.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 4 - π}{4}$

Step 3.2.6.3

Simplify the numerator.

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Step 3.2.6.3.1

Multiply $4$ by $2$ .

$x = \frac{8 π - π}{4}$

Step 3.2.6.3.2

Subtract $π$ from $8 π$ .

$x = \frac{7 π}{4}$

Step 3.2.7

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

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Step 3.2.7.1

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

$\frac{2 π}{| b |}$

Step 3.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 3.2.7.3

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

$\frac{2 π}{1}$

Step 3.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 3.2.8

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

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n$ , for any integer $n$

Step 4

Set $\sin (x) - 1$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $\sin (x) - 1$ equal to $0$ .

$\sin (x) - 1 = 0$

Step 4.2

Solve $\sin (x) - 1 = 0$ for $x$ .

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Step 4.2.1

Add $1$ to both sides of the equation.

$\sin (x) = 1$

Step 4.2.2

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

$x = \arcsin (1)$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 4.2.4

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.

$x = π - \frac{π}{2}$

Step 4.2.5

Simplify $π - \frac{π}{2}$ .

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Step 4.2.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.2.5.2

Combine fractions.

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Step 4.2.5.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 4.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 4.2.5.3

Simplify the numerator.

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Step 4.2.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 4.2.5.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 4.2.6

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

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Step 4.2.6.1

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

$\frac{2 π}{| b |}$

Step 4.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 4.2.6.3

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

$\frac{2 π}{1}$

Step 4.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 4.2.7

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

$x = \frac{π}{2} + 2 π n$ , for any integer $n$

Step 5

The final solution is all the values that make $(\sqrt{2} \cos (x) - 1) (\sin (x) - 1) = 0$ true.

$x = \frac{π}{4} + 2 π n, \frac{7 π}{4} + 2 π n, \frac{π}{2} + 2 π n$ , for any integer $n$