Trigonometry Examples

$3 \sin (x) - 3 \cos (x) = 1$

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 $θ$ .

$\tan^{-1} (- \frac{3}{3})$

Step 3

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

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Step 3.1

Cancel the common factor of $3$ .

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Step 3.1.1

Cancel the common factor.

$θ = \tan^{-1} (- \frac{3}{3})$

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

Raise $3$ to the power of $2$ .

$R = \sqrt{9 + {(- 3)}^{2}}$

Step 4.2

Raise $- 3$ to the power of $2$ .

$R = \sqrt{9 + 9}$

Step 4.3

Add $9$ and $9$ .

$R = \sqrt{18}$

Step 4.4

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.4.1

Factor $9$ out of $18$ .

$R = \sqrt{9 (2)}$

Step 4.4.2

Rewrite $9$ as $3^{2}$ .

$R = \sqrt{3^{2} \cdot 2}$

Step 4.5

Pull terms out from under the radical.

$R = 3 \sqrt{2}$

Step 5

Substitute the known values into the equation.

$(3 \sqrt{2}) \sin (x - \frac{π}{4}) = 1$

Step 6

Divide each term in $3 \sqrt{2} \sin (x - \frac{π}{4}) = 1$ by $3 \sqrt{2}$ and simplify.

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Step 6.1

Divide each term in $3 \sqrt{2} \sin (x - \frac{π}{4}) = 1$ by $3 \sqrt{2}$ .

$\frac{3 \sqrt{2} \sin (x - \frac{π}{4})}{3 \sqrt{2}} = \frac{1}{3 \sqrt{2}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $3$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{3 \sqrt{2} \sin (x - \frac{π}{4})}{3 \sqrt{2}} = \frac{1}{3 \sqrt{2}}$

Step 6.2.1.2

Rewrite the expression.

$\frac{\sqrt{2} \sin (x - \frac{π}{4})}{\sqrt{2}} = \frac{1}{3 \sqrt{2}}$

Step 6.2.2

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

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Step 6.2.2.1

Cancel the common factor.

$\frac{\sqrt{2} \sin (x - \frac{π}{4})}{\sqrt{2}} = \frac{1}{3 \sqrt{2}}$

Step 6.2.2.2

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

$\sin (x - \frac{π}{4}) = \frac{1}{3 \sqrt{2}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply $\frac{1}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x - \frac{π}{4}) = \frac{1}{3 \sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply $\frac{1}{3 \sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 \sqrt{2} \sqrt{2}}$

Step 6.3.2.2

Move $\sqrt{2}$ .

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 (\sqrt{2} \sqrt{2})}$

Step 6.3.2.3

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 ({\sqrt{2}}^{1} \sqrt{2})}$

Step 6.3.2.4

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}$

Step 6.3.2.5

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 {\sqrt{2}}^{1 + 1}}$

Step 6.3.2.6

Add $1$ and $1$ .

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 {\sqrt{2}}^{2}}$

Step 6.3.2.7

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

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Step 6.3.2.7.1

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 {(2^{\frac{1}{2}})}^{2}}$

Step 6.3.2.7.2

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.7.3

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

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 6.3.2.7.4

Cancel the common factor of $2$ .

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Step 6.3.2.7.4.1

Cancel the common factor.

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

Step 6.3.2.7.4.2

Rewrite the expression.

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 \cdot 2^{1}}$

Step 6.3.2.7.5

Evaluate the exponent.

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{3 \cdot 2}$

Step 6.3.3

Multiply $3$ by $2$ .

$\sin (x - \frac{π}{4}) = \frac{\sqrt{2}}{6}$

Step 7

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

$x - \frac{π}{4} = \sin^{-1} (\frac{\sqrt{2}}{6})$

Step 8

Simplify the right side.

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Step 8.1

Evaluate $\sin^{-1} (\frac{\sqrt{2}}{6})$ .

$x - \frac{π}{4} = 0.23794112$

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.

$x = 0.23794112 + \frac{π}{4}$

Step 9.2

Add $0.23794112$ and $\frac{π}{4}$ .

$x = 1.02333928$

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.

$x - \frac{π}{4} = (3.14159265) - 0.23794112$

Step 11

Solve for $x$ .

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Step 11.1

Subtract $0.23794112$ from $3.14159265$ .

$x - \frac{π}{4} = 2.90365152$

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.

$x = 2.90365152 + \frac{π}{4}$

Step 11.2.2

Add $2.90365152$ and $\frac{π}{4}$ .

$x = 3.68904969$

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.

$x = 1.02333928 + 2 π n, 3.68904969 + 2 π n$ , for any integer $n$