Solve for x sin(x)+cos(x) = square root of 2

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

Square both sides of the equation.

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

Rewrite ${(\sin (x) + \cos (x))}^{2}$ as $(\sin (x) + \cos (x)) (\sin (x) + \cos (x))$ .

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

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

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Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

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Simplify each term.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $\cos (x) \cos (x)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Reorder the factors of $\sin (x) \cos (x)$ .

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

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

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

Move $\cos^{2} (x)$ .

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

Apply pythagorean identity.

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

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

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Rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

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

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

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

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

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $1$ by $1$ .

$1 + 2 \cos (x) \sin (x) = 2$

Evaluate the exponent.

$1 + 2 \cos (x) \sin (x) = 2$

Simplify each term.

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

$1 + \sin (x) (2 \cos (x)) = 2$

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

$1 + 2 \cdot (\sin (x) \cos (x)) = 2$

Apply the sine double–angle identity.

$1 + \sin (2 x) = 2$

Move all terms not containing $\sin (2 x)$ to the right side of the equation.

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Subtract $1$ from both sides of the equation.

$\sin (2 x) = 2 - 1$

Subtract $1$ from $2$ .

$\sin (2 x) = 1$

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

$2 x = \arcsin (1)$

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

$2 x = \frac{π}{2}$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2} \cdot \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{π}{2} \cdot \frac{1}{2}$

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{π}{2}$ and $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

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.

$2 x = π - \frac{π}{2}$

Simplify the expression to find the second solution.

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To write $\frac{π}{1}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x = \frac{π}{1} \cdot \frac{2}{2} - \frac{π}{2}$

Write each expression with a common denominator of $2$ , by multiplying each by an appropriate factor of $1$ .

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Combine.

$\frac{π \cdot 2}{1 \cdot 2} - \frac{π}{2}$

Multiply $2$ by $1$ .

$\frac{π \cdot 2}{2} - \frac{π}{2}$

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

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 2 - π}{2}$

Subtract $π$ from $π \cdot 2$ .

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Reorder $π$ and $2$ .

$\frac{2 \cdot π - π}{2}$

Subtract $π$ from $2 \cdot π$ .

$\frac{π}{2}$

$2 x = \frac{π}{2}$

Divide each term by $2$ and simplify.

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Divide each term in $2 x = \frac{π}{2}$ by $2$ .

$\frac{2 x}{2} = \frac{π}{2} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 x}{2} = \frac{π}{2} \cdot \frac{1}{2}$

Divide $x$ by $1$ .

$x = \frac{π}{2} \cdot \frac{1}{2}$

Multiply $\frac{π}{2} \cdot \frac{1}{2}$ .

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Multiply $\frac{π}{2}$ and $\frac{1}{2}$ .

$x = \frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Find the period.

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

$\frac{2 π}{| b |}$

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Solve the equation.

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The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

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

Verify each of the solutions by substituting them into $\sin (x) + \cos (x) = \sqrt{2}$ and solving.

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

Solve for x sin(x)+cos(x) = square root of 2

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