Trigonometry Examples

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

Step 1

Square both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

Apply the distributive property.

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

Step 2.2.2

Apply the distributive property.

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

Step 2.2.3

Apply the distributive property.

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

Step 2.3

Simplify and combine like terms.

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Step 2.3.1

Simplify each term.

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Step 2.3.1.1

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

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Step 2.3.1.1.1

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

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

Step 2.3.1.1.2

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

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

Step 2.3.1.1.3

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

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

Step 2.3.1.1.4

Add $1$ and $1$ .

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

Step 2.3.1.2

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

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Step 2.3.1.2.1

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

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

Step 2.3.1.2.2

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

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

Step 2.3.1.2.3

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

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

Step 2.3.1.2.4

Add $1$ and $1$ .

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

Step 2.3.2

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

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

Step 2.3.3

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

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

Step 2.4

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

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

Step 2.5

Apply pythagorean identity.

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

Step 2.6

Simplify each term.

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

Apply the sine double-angle identity.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Evaluate the exponent.

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

Step 4

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

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Step 4.1

Subtract $1$ from both sides of the equation.

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

Step 4.2

Subtract $1$ from $2$ .

$\sin (2 x) = 1$

Step 5

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

$2 x = \arcsin (1)$

Step 6

Simplify the right side.

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Step 6.1

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

$2 x = \frac{π}{2}$

Step 7

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

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Step 7.1

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

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

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $2$ .

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Step 7.2.1.1

Cancel the common factor.

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

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 7.3.2

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

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Step 7.3.2.1

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

$x = \frac{π}{2 \cdot 2}$

Step 7.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 8

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.

$2 x = π - \frac{π}{2}$

Step 9

Solve for $x$ .

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Step 9.1

Simplify.

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Step 9.1.1

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

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

Step 9.1.2

Combine $π$ and $\frac{2}{2}$ .

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

Step 9.1.3

Combine the numerators over the common denominator.

$2 x = \frac{π \cdot 2 - π}{2}$

Step 9.1.4

Subtract $π$ from $π \cdot 2$ .

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Step 9.1.4.1

Reorder $π$ and $2$ .

$2 x = \frac{2 \cdot π - π}{2}$

Step 9.1.4.2

Subtract $π$ from $2 \cdot π$ .

$2 x = \frac{π}{2}$

Step 9.2

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

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Step 9.2.1

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

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

Step 9.2.2

Simplify the left side.

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Step 9.2.2.1

Cancel the common factor of $2$ .

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Step 9.2.2.1.1

Cancel the common factor.

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

Step 9.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{2}$

Step 9.2.3

Simplify the right side.

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Step 9.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{2}$

Step 9.2.3.2

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

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Step 9.2.3.2.1

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

$x = \frac{π}{2 \cdot 2}$

Step 9.2.3.2.2

Multiply $2$ by $2$ .

$x = \frac{π}{4}$

Step 10

Find the period of $\sin (2 x)$ .

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Step 10.1

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

$\frac{2 π}{| b |}$

Step 10.2

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

$\frac{2 π}{| 2 |}$

Step 10.3

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

$\frac{2 π}{2}$

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 10.4.2

Divide $π$ by $1$ .

$π$

Step 11

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 $n$

Step 12

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 $n$