Trigonometry Examples

$\sin (θ) - \cos (θ) = \sqrt{2}$

Step 1

Use the identity to solve the equation. In this identity, $θ$ represents the angle created by plotting point $(a, b)$ on a graph and therefore can be found using $θ = \tan^{-1} (\frac{b}{a})$ .

$a \sin (x) + b \cos (x) = R \sin (x + θ)$ where $R = \sqrt{a^{2} + b^{2}}$ and $θ = \tan^{-1} (\frac{b}{a})$

Step 2

Set up the equation to find the value of $θ$ .

$\tan^{-1} (- \frac{1}{1})$

Step 3

Take the inverse tangent to solve the equation for $θ$ .

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Step 3.1

Cancel the common factor of $1$ .

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Step 3.1.1

Cancel the common factor.

$θ = \tan^{-1} (- \frac{1}{1})$

Step 3.1.2

Rewrite the expression.

$θ = \tan^{-1} (- 1 \cdot 1)$

Step 3.2

Multiply $- 1$ by $1$ .

$θ = \tan^{-1} (- 1)$

Step 3.3

The exact value of $\tan^{-1} (- 1)$ is $- \frac{π}{4}$ .

$θ = - \frac{π}{4}$

Step 4

Solve to find the value of $R$ .

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Step 4.1

One to any power is one.

$R = \sqrt{1 + {(- 1)}^{2}}$

Step 4.2

Raise $- 1$ to the power of $2$ .

$R = \sqrt{1 + 1}$

Step 4.3

Add $1$ and $1$ .

$R = \sqrt{2}$

Step 5

Substitute the known values into the equation.

$(\sqrt{2}) \sin (θ - \frac{π}{4}) = \sqrt{2}$

Step 6

Divide each term in $\sqrt{2} \sin (θ - \frac{π}{4}) = \sqrt{2}$ by $\sqrt{2}$ and simplify.

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Step 6.1

Divide each term in $\sqrt{2} \sin (θ - \frac{π}{4}) = \sqrt{2}$ by $\sqrt{2}$ .

$\frac{\sqrt{2} \sin (θ - \frac{π}{4})}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

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

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Step 6.2.1.1

Cancel the common factor.

$\frac{\sqrt{2} \sin (θ - \frac{π}{4})}{\sqrt{2}} = \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.2.1.2

Divide $\sin (θ - \frac{π}{4})$ by $1$ .

$\sin (θ - \frac{π}{4}) = \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

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Step 6.3.1.1

Cancel the common factor.

$\sin (θ - \frac{π}{4}) = \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3.1.2

Rewrite the expression.

$\sin (θ - \frac{π}{4}) = 1$

Step 7

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

$θ - \frac{π}{4} = \sin^{-1} (1)$

Step 8

Simplify the right side.

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Step 8.1

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

$θ - \frac{π}{4} = \frac{π}{2}$

Step 9

Move all terms not containing $θ$ to the right side of the equation.

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Step 9.1

Add $\frac{π}{4}$ to both sides of the equation.

$θ = \frac{π}{2} + \frac{π}{4}$

Step 9.2

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

$θ = \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4}$

Step 9.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 9.3.1

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

$θ = \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4}$

Step 9.3.2

Multiply $2$ by $2$ .

$θ = \frac{π \cdot 2}{4} + \frac{π}{4}$

Step 9.4

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 2 + π}{4}$

Step 9.5

Simplify the numerator.

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Step 9.5.1

Move $2$ to the left of $π$ .

$θ = \frac{2 \cdot π + π}{4}$

Step 9.5.2

Add $2 π$ and $π$ .

$θ = \frac{3 π}{4}$

Step 10

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.

$θ - \frac{π}{4} = π - \frac{π}{2}$

Step 11

Solve for $θ$ .

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Step 11.1

Simplify $π - \frac{π}{2}$ .

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Step 11.1.1

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

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

Step 11.1.2

Combine fractions.

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Step 11.1.2.1

Combine $π$ and $\frac{2}{2}$ .

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

Step 11.1.2.2

Combine the numerators over the common denominator.

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

Step 11.1.3

Simplify the numerator.

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Step 11.1.3.1

Move $2$ to the left of $π$ .

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

Step 11.1.3.2

Subtract $π$ from $2 π$ .

$θ - \frac{π}{4} = \frac{π}{2}$

Step 11.2

Move all terms not containing $θ$ to the right side of the equation.

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Step 11.2.1

Add $\frac{π}{4}$ to both sides of the equation.

$θ = \frac{π}{2} + \frac{π}{4}$

Step 11.2.2

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

$θ = \frac{π}{2} \cdot \frac{2}{2} + \frac{π}{4}$

Step 11.2.3

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 11.2.3.1

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

$θ = \frac{π \cdot 2}{2 \cdot 2} + \frac{π}{4}$

Step 11.2.3.2

Multiply $2$ by $2$ .

$θ = \frac{π \cdot 2}{4} + \frac{π}{4}$

Step 11.2.4

Combine the numerators over the common denominator.

$θ = \frac{π \cdot 2 + π}{4}$

Step 11.2.5

Simplify the numerator.

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Step 11.2.5.1

Move $2$ to the left of $π$ .

$θ = \frac{2 \cdot π + π}{4}$

Step 11.2.5.2

Add $2 π$ and $π$ .

$θ = \frac{3 π}{4}$

Step 12

Find the period of $\sin (θ - \frac{π}{4})$ .

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Step 12.1

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

$\frac{2 π}{| b |}$

Step 12.2

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

$\frac{2 π}{| 1 |}$

Step 12.3

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

$\frac{2 π}{1}$

Step 12.4

Divide $2 π$ by $1$ .

$2 π$

Step 13

The period of the $\sin (θ - \frac{π}{4})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$θ = \frac{3 π}{4} + 2 π n$ , for any integer $n$