Trigonometry Examples

$\sin (x) \cos (\frac{π}{7}) - \sin (\frac{π}{7}) \cos (x) = \frac{\sqrt{2}}{2}$

Step 1

Simplify each term.

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Step 1.1

Evaluate $\cos (\frac{π}{7})$ .

$\sin (x) \cdot 0.90096886 - \sin (\frac{π}{7}) \cos (x) = \frac{\sqrt{2}}{2}$

Step 1.2

Move $0.90096886$ to the left of $\sin (x)$ .

$0.90096886 \cdot \sin (x) - \sin (\frac{π}{7}) \cos (x) = \frac{\sqrt{2}}{2}$

Step 1.3

Evaluate $\sin (\frac{π}{7})$ .

$0.90096886 \sin (x) - 1 \cdot (0.43388373 \cos (x)) = \frac{\sqrt{2}}{2}$

Step 1.4

Multiply $- 1$ by $0.43388373$ .

$0.90096886 \sin (x) - 0.43388373 \cos (x) = \frac{\sqrt{2}}{2}$

Step 2

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 3

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

$\tan^{-1} (- \frac{0.43388373}{0.90096886})$

Step 4

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

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Step 4.1

Divide $0.43388373$ by $0.90096886$ .

$θ = \tan^{-1} (- 1 \cdot 0.48157461)$

Step 4.2

Multiply $- 1$ by $0.48157461$ .

$θ = \tan^{-1} (- 0.48157461)$

Step 4.3

Evaluate $\tan^{-1} (- 0.48157461)$ .

$θ = - \frac{π}{7}$

Step 5

Solve to find the value of $R$ .

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Step 5.1

Raise $0.90096886$ to the power of $2$ .

$R = \sqrt{0.8117449 + {(- 0.43388373)}^{2}}$

Step 5.2

Raise $- 0.43388373$ to the power of $2$ .

$R = \sqrt{0.8117449 + 0.18825509}$

Step 5.3

Add $0.8117449$ and $0.18825509$ .

$R = \sqrt{1}$

Step 6

Substitute the known values into the equation.

$(\sqrt{1}) \sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2}$

Step 7

Divide each term in $\sqrt{1} \sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2}$ by $\sqrt{1}$ and simplify.

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Step 7.1

Divide each term in $\sqrt{1} \sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2}$ by $\sqrt{1}$ .

$\frac{\sqrt{1} \sin (x - \frac{π}{7})}{\sqrt{1}} = \frac{\frac{\sqrt{2}}{2}}{\sqrt{1}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $\sqrt{1}$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{\sqrt{1} \sin (x - \frac{π}{7})}{\sqrt{1}} = \frac{\frac{\sqrt{2}}{2}}{\sqrt{1}}$

Step 7.2.1.2

Divide $\sin (x - \frac{π}{7})$ by $1$ .

$\sin (x - \frac{π}{7}) = \frac{\frac{\sqrt{2}}{2}}{\sqrt{1}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Multiply the numerator by the reciprocal of the denominator.

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{1}{\sqrt{1}}$

Step 7.3.2

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} (\frac{1}{\sqrt{1}} \cdot \frac{\sqrt{1}}{\sqrt{1}})$

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{\sqrt{1} \sqrt{1}}$

Step 7.3.3.2

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{1} \sqrt{1}}$

Step 7.3.3.3

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{1} {\sqrt{1}}^{1}}$

Step 7.3.3.4

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{1 + 1}}$

Step 7.3.3.5

Add $1$ and $1$ .

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{{\sqrt{1}}^{2}}$

Step 7.3.3.6

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

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Step 7.3.3.6.1

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{{(1^{\frac{1}{2}})}^{2}}$

Step 7.3.3.6.2

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{1^{\frac{1}{2} \cdot 2}}$

Step 7.3.3.6.3

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{1^{\frac{2}{2}}}$

Step 7.3.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.3.6.4.1

Cancel the common factor.

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{1^{\frac{2}{2}}}$

Step 7.3.3.6.4.2

Rewrite the expression.

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{1^{1}}$

Step 7.3.3.6.5

Evaluate the exponent.

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{1}}{1}$

Step 7.3.4

Evaluate the root.

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot \frac{1}{1}$

Step 7.3.5

Divide $1$ by $1$ .

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2} \cdot 1$

Step 7.3.6

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

$\sin (x - \frac{π}{7}) = \frac{\sqrt{2}}{2}$

Step 8

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

$x - \frac{π}{7} = \sin^{-1} (\frac{\sqrt{2}}{2})$

Step 9

Simplify the right side.

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Step 9.1

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

$x - \frac{π}{7} = \frac{π}{4}$

Step 10

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

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Step 10.1

Add $\frac{π}{7}$ to both sides of the equation.

$x = \frac{π}{4} + \frac{π}{7}$

Step 10.2

To write $\frac{π}{4}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \frac{π}{4} \cdot \frac{7}{7} + \frac{π}{7}$

Step 10.3

To write $\frac{π}{7}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{π}{4} \cdot \frac{7}{7} + \frac{π}{7} \cdot \frac{4}{4}$

Step 10.4

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

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Step 10.4.1

Multiply $\frac{π}{4}$ by $\frac{7}{7}$ .

$x = \frac{π \cdot 7}{4 \cdot 7} + \frac{π}{7} \cdot \frac{4}{4}$

Step 10.4.2

Multiply $4$ by $7$ .

$x = \frac{π \cdot 7}{28} + \frac{π}{7} \cdot \frac{4}{4}$

Step 10.4.3

Multiply $\frac{π}{7}$ by $\frac{4}{4}$ .

$x = \frac{π \cdot 7}{28} + \frac{π \cdot 4}{7 \cdot 4}$

Step 10.4.4

Multiply $7$ by $4$ .

$x = \frac{π \cdot 7}{28} + \frac{π \cdot 4}{28}$

Step 10.5

Combine the numerators over the common denominator.

$x = \frac{π \cdot 7 + π \cdot 4}{28}$

Step 10.6

Simplify the numerator.

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Step 10.6.1

Move $7$ to the left of $π$ .

$x = \frac{7 \cdot π + π \cdot 4}{28}$

Step 10.6.2

Move $4$ to the left of $π$ .

$x = \frac{7 π + 4 \cdot π}{28}$

Step 10.6.3

Add $7 π$ and $4 π$ .

$x = \frac{11 π}{28}$

Step 11

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.

$x - \frac{π}{7} = π - \frac{π}{4}$

Step 12

Solve for $x$ .

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Step 12.1

Simplify $π - \frac{π}{4}$ .

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Step 12.1.1

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

$x - \frac{π}{7} = π \cdot \frac{4}{4} - \frac{π}{4}$

Step 12.1.2

Combine fractions.

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Step 12.1.2.1

Combine $π$ and $\frac{4}{4}$ .

$x - \frac{π}{7} = \frac{π \cdot 4}{4} - \frac{π}{4}$

Step 12.1.2.2

Combine the numerators over the common denominator.

$x - \frac{π}{7} = \frac{π \cdot 4 - π}{4}$

Step 12.1.3

Simplify the numerator.

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Step 12.1.3.1

Move $4$ to the left of $π$ .

$x - \frac{π}{7} = \frac{4 \cdot π - π}{4}$

Step 12.1.3.2

Subtract $π$ from $4 π$ .

$x - \frac{π}{7} = \frac{3 π}{4}$

Step 12.2

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

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Step 12.2.1

Add $\frac{π}{7}$ to both sides of the equation.

$x = \frac{3 π}{4} + \frac{π}{7}$

Step 12.2.2

To write $\frac{3 π}{4}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$x = \frac{3 π}{4} \cdot \frac{7}{7} + \frac{π}{7}$

Step 12.2.3

To write $\frac{π}{7}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x = \frac{3 π}{4} \cdot \frac{7}{7} + \frac{π}{7} \cdot \frac{4}{4}$

Step 12.2.4

Write each expression with a common denominator of $28$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.2.4.1

Multiply $\frac{3 π}{4}$ by $\frac{7}{7}$ .

$x = \frac{3 π \cdot 7}{4 \cdot 7} + \frac{π}{7} \cdot \frac{4}{4}$

Step 12.2.4.2

Multiply $4$ by $7$ .

$x = \frac{3 π \cdot 7}{28} + \frac{π}{7} \cdot \frac{4}{4}$

Step 12.2.4.3

Multiply $\frac{π}{7}$ by $\frac{4}{4}$ .

$x = \frac{3 π \cdot 7}{28} + \frac{π \cdot 4}{7 \cdot 4}$

Step 12.2.4.4

Multiply $7$ by $4$ .

$x = \frac{3 π \cdot 7}{28} + \frac{π \cdot 4}{28}$

Step 12.2.5

Combine the numerators over the common denominator.

$x = \frac{3 π \cdot 7 + π \cdot 4}{28}$

Step 12.2.6

Simplify the numerator.

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Step 12.2.6.1

Multiply $7$ by $3$ .

$x = \frac{21 π + π \cdot 4}{28}$

Step 12.2.6.2

Move $4$ to the left of $π$ .

$x = \frac{21 π + 4 \cdot π}{28}$

Step 12.2.6.3

Add $21 π$ and $4 π$ .

$x = \frac{25 π}{28}$

Step 13

Find the period of $\sin (x - \frac{π}{7})$ .

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Step 13.1

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

$\frac{2 π}{| b |}$

Step 13.2

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

$\frac{2 π}{| 1 |}$

Step 13.3

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

$\frac{2 π}{1}$

Step 13.4

Divide $2 π$ by $1$ .

$2 π$

Step 14

The period of the $\sin (x - \frac{π}{7})$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = \frac{11 π}{28} + 2 π n, \frac{25 π}{28} + 2 π n$ , for any integer $n$