Precalculus Examples

$\sin (x) + 3 \cos (x) = 2$

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 $θ$ .

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

Step 3

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

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Step 3.1

Divide $3$ by $1$ .

$θ = \tan^{-1} (3)$

Step 3.2

Evaluate $\tan^{-1} (3)$ .

$θ = 1.24904577$

Step 4

Solve to find the value of $R$ .

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Step 4.1

One to any power is one.

$R = \sqrt{1 + {(3)}^{2}}$

Step 4.2

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

$R = \sqrt{1 + 9}$

Step 4.3

Add $1$ and $9$ .

$R = \sqrt{10}$

Step 5

Substitute the known values into the equation.

$(\sqrt{10}) \sin (x + 1.24904577) = 2$

Step 6

Divide each term in $\sqrt{10} \sin (x + 1.24904577) = 2$ by $\sqrt{10}$ and simplify.

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Step 6.1

Divide each term in $\sqrt{10} \sin (x + 1.24904577) = 2$ by $\sqrt{10}$ .

$\frac{\sqrt{10} \sin (x + 1.24904577)}{\sqrt{10}} = \frac{2}{\sqrt{10}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\sqrt{10}$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{\sqrt{10} \sin (x + 1.24904577)}{\sqrt{10}} = \frac{2}{\sqrt{10}}$

Step 6.2.1.2

Divide $\sin (x + 1.24904577)$ by $1$ .

$\sin (x + 1.24904577) = \frac{2}{\sqrt{10}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply $\frac{2}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (x + 1.24904577) = \frac{2}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply $\frac{2}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 6.3.2.2

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 6.3.2.3

Raise $\sqrt{10}$ to the power of $1$ .

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 6.3.2.4

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

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 6.3.2.5

Add $1$ and $1$ .

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 6.3.2.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 6.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

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

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

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

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{10^{\frac{2}{2}}}$

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.

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{10^{\frac{2}{2}}}$

Step 6.3.2.6.4.2

Rewrite the expression.

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{10^{1}}$

Step 6.3.2.6.5

Evaluate the exponent.

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{10}$

Step 6.3.3

Cancel the common factor of $2$ and $10$ .

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Step 6.3.3.1

Factor $2$ out of $2 \sqrt{10}$ .

$\sin (x + 1.24904577) = \frac{2 (\sqrt{10})}{10}$

Step 6.3.3.2

Cancel the common factors.

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Step 6.3.3.2.1

Factor $2$ out of $10$ .

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{2 \cdot 5}$

Step 6.3.3.2.2

Cancel the common factor.

$\sin (x + 1.24904577) = \frac{2 \sqrt{10}}{2 \cdot 5}$

Step 6.3.3.2.3

Rewrite the expression.

$\sin (x + 1.24904577) = \frac{\sqrt{10}}{5}$

Step 7

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

$x + 1.24904577 = \sin^{-1} (\frac{\sqrt{10}}{5})$

Step 8

Simplify the right side.

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Step 8.1

Evaluate $\sin^{-1} (\frac{\sqrt{10}}{5})$ .

$x + 1.24904577 = 0.6847192$

Step 9

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

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Step 9.1

Subtract $1.24904577$ from both sides of the equation.

$x = 0.6847192 - 1.24904577$

Step 9.2

Subtract $1.24904577$ from $0.6847192$ .

$x = - 0.56432656$

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.

$x + 1.24904577 = (3.14159265) - 0.6847192$

Step 11

Solve for $x$ .

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Step 11.1

Subtract $0.6847192$ from $3.14159265$ .

$x + 1.24904577 = 2.45687345$

Step 11.2

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

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Step 11.2.1

Subtract $1.24904577$ from both sides of the equation.

$x = 2.45687345 - 1.24904577$

Step 11.2.2

Subtract $1.24904577$ from $2.45687345$ .

$x = 1.20782767$

Step 12

Find the period of $\sin (x + 1.24904577)$ .

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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

Add $2 π$ to every negative angle to get positive angles.

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Step 13.1

Add $2 π$ to $- 0.56432656$ to find the positive angle.

$- 0.56432656 + 2 π$

Step 13.2

Subtract $0.56432656$ from $2 π$ .

$5.71885873$

Step 13.3

List the new angles.

$x = 5.71885873$

Step 14

The period of the $\sin (x + 1.24904577)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 1.20782767 + 2 π n, 5.71885873 + 2 π n$ , for any integer $n$