Find All Complex Solutions sin(theta)+cos(theta) = square root of 2

$\sin (θ) + \cos (θ) = \sqrt{2}$

Square both sides of the equation.

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

Simplify ${(\sin (θ) + \cos (θ))}^{2}$ .

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $\sin (θ) \sin (θ)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

Multiply $\cos (θ) \cos (θ)$ .

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

Move $\cos^{2} (θ)$ .

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

Apply pythagorean identity.

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

Simplify each term.

Reorder $2 \cos (θ)$ and $\sin (θ)$ .

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

Reorder $\sin (θ)$ and $2$ .

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

Apply the sine double-angle identity.

$1 + \sin (2 θ) = {(\sqrt{2})}^{2}$

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

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

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

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

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

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

$1 + \sin (2 θ) = 2^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$1 + \sin (2 θ) = 2^{\frac{2}{2}}$

Rewrite the expression.

$1 + \sin (2 θ) = 2^{1}$

Evaluate the exponent.

$1 + \sin (2 θ) = 2$

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

Subtract $1$ from both sides of the equation.

$\sin (2 θ) = 2 - 1$

Subtract $1$ from $2$ .

$\sin (2 θ) = 1$

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

$2 θ = \arcsin (1)$

Simplify the right side.

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

$2 θ = \frac{π}{2}$

Divide each term in $2 θ = \frac{π}{2}$ by $2$ and simplify.

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

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{2} \cdot \frac{1}{2}$

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

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

$θ = \frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$θ = \frac{π}{4}$

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.

$2 θ = π - \frac{π}{2}$

Solve for $θ$ .

Simplify.

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

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

Combine $π$ and $\frac{2}{2}$ .

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

Combine the numerators over the common denominator.

$2 θ = \frac{π \cdot 2 - π}{2}$

Subtract $π$ from $π \cdot 2$ .

Reorder $π$ and $2$ .

$2 θ = \frac{2 \cdot π - π}{2}$

Subtract $π$ from $2 \cdot π$ .

$2 θ = \frac{π}{2}$

Divide each term in $2 θ = \frac{π}{2}$ by $2$ and simplify.

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

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 θ}{2} = \frac{\frac{π}{2}}{2}$

Divide $θ$ by $1$ .

$θ = \frac{\frac{π}{2}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$θ = \frac{π}{2} \cdot \frac{1}{2}$

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

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

$θ = \frac{π}{2 \cdot 2}$

Multiply $2$ by $2$ .

$θ = \frac{π}{4}$

Find the period of $\sin (2 θ)$ .

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.

$\frac{2 π}{| 2 |}$

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 $2$ .

Cancel the common factor.

$\frac{2 π}{2}$

Divide $π$ by $1$ .

$π$

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

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

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

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

Find All Complex Solutions sin(theta)+cos(theta) = square root of 2

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