Trigonometry Examples

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

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.

$(\sqrt{2}) \sin (x - \frac{π}{4}) = 1$

Step 6

Divide each term in $\sqrt{2} \sin (x - \frac{π}{4}) = 1$ by $\sqrt{2}$ and simplify.

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Step 6.1

Divide each term in $\sqrt{2} \sin (x - \frac{π}{4}) = 1$ by $\sqrt{2}$ .

$\frac{\sqrt{2} \sin (x - \frac{π}{4})}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

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.

$\frac{\sqrt{2} \sin (x - \frac{π}{4})}{\sqrt{2}} = \frac{1}{\sqrt{2}}$

Step 6.2.1.2

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

$\sin (x - \frac{π}{4}) = \frac{1}{\sqrt{2}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

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

$\sin (x - \frac{π}{4}) = \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

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

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

Step 6.3.2.2

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}$

Step 6.3.2.3

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}$

Step 6.3.2.4

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 6.3.2.5

Add $1$ and $1$ .

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

Step 6.3.2.6

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

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Step 6.3.2.6.1

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

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

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

Step 6.3.2.6.4

Cancel the common factor of $2$ .

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Step 6.3.2.6.4.1

Cancel the common factor.

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

Step 6.3.2.6.4.2

Rewrite the expression.

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{2^{1}}$

Step 6.3.2.6.5

Evaluate the exponent.

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

Step 7

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

$x - \frac{π}{4} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Step 8

Simplify the right side.

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Step 8.1

The exact value of $\sin^{-1} (\frac{\sqrt{2}}{2})$ is $\frac{π}{4}$ .

$x - \frac{π}{4} = \frac{π}{4}$

Step 9

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

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Step 9.1

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

$x = \frac{π}{4} + \frac{π}{4}$

Step 9.2

Combine the numerators over the common denominator.

$x = \frac{π + π}{4}$

Step 9.3

Add $π$ and $π$ .

$x = \frac{2 π}{4}$

Step 9.4

Cancel the common factor of $2$ and $4$ .

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Step 9.4.1

Factor $2$ out of $2 π$ .

$x = \frac{2 (π)}{4}$

Step 9.4.2

Cancel the common factors.

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Step 9.4.2.1

Factor $2$ out of $4$ .

$x = \frac{2 π}{2 \cdot 2}$

Step 9.4.2.2

Cancel the common factor.

$x = \frac{2 π}{2 \cdot 2}$

Step 9.4.2.3

Rewrite the expression.

$x = \frac{π}{2}$

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.

$x - \frac{π}{4} = π - \frac{π}{4}$

Step 11

Solve for $x$ .

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Step 11.1

Simplify $π - \frac{π}{4}$ .

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Step 11.1.1

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

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

Step 11.1.2

Combine fractions.

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Step 11.1.2.1

Combine $π$ and $\frac{4}{4}$ .

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

Step 11.1.2.2

Combine the numerators over the common denominator.

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

Step 11.1.3

Simplify the numerator.

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Step 11.1.3.1

Move $4$ to the left of $π$ .

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

Step 11.1.3.2

Subtract $π$ from $4 π$ .

$x - \frac{π}{4} = \frac{3 π}{4}$

Step 11.2

Move all terms not containing $x$ 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.

$x = \frac{3 π}{4} + \frac{π}{4}$

Step 11.2.2

Combine the numerators over the common denominator.

$x = \frac{3 π + π}{4}$

Step 11.2.3

Add $3 π$ and $π$ .

$x = \frac{4 π}{4}$

Step 11.2.4

Cancel the common factor of $4$ .

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Step 11.2.4.1

Cancel the common factor.

$x = \frac{4 π}{4}$

Step 11.2.4.2

Divide $π$ by $1$ .

$x = π$

Step 12

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

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

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