Trigonometry Examples

$5 \sin (x) + 2 \cos (x) = 1$

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 $θ$ .

$\tan^{-1} (\frac{2}{5})$

Step 3

Evaluate $\tan^{-1} (\frac{2}{5})$ .

$θ = 0.38050637$

Step 4

Solve to find the value of $R$ .

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Step 4.1

Raise $5$ to the power of $2$ .

$R = \sqrt{25 + {(2)}^{2}}$

Step 4.2

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

$R = \sqrt{25 + 4}$

Step 4.3

Add $25$ and $4$ .

$R = \sqrt{29}$

Step 5

Substitute the known values into the equation.

$(\sqrt{29}) \sin (x + 0.38050637) = 1$

Step 6

Divide each term in $\sqrt{29} \sin (x + 0.38050637) = 1$ by $\sqrt{29}$ and simplify.

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Step 6.1

Divide each term in $\sqrt{29} \sin (x + 0.38050637) = 1$ by $\sqrt{29}$ .

$\frac{\sqrt{29} \sin (x + 0.38050637)}{\sqrt{29}} = \frac{1}{\sqrt{29}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\sqrt{29}$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{\sqrt{29} \sin (x + 0.38050637)}{\sqrt{29}} = \frac{1}{\sqrt{29}}$

Step 6.2.1.2

Divide $\sin (x + 0.38050637)$ by $1$ .

$\sin (x + 0.38050637) = \frac{1}{\sqrt{29}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply $\frac{1}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sin (x + 0.38050637) = \frac{1}{\sqrt{29}} \cdot \frac{\sqrt{29}}{\sqrt{29}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply $\frac{1}{\sqrt{29}}$ by $\frac{\sqrt{29}}{\sqrt{29}}$ .

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{\sqrt{29} \sqrt{29}}$

Step 6.3.2.2

Raise $\sqrt{29}$ to the power of $1$ .

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{{\sqrt{29}}^{1} \sqrt{29}}$

Step 6.3.2.3

Raise $\sqrt{29}$ to the power of $1$ .

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{{\sqrt{29}}^{1} {\sqrt{29}}^{1}}$

Step 6.3.2.4

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

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{{\sqrt{29}}^{1 + 1}}$

Step 6.3.2.5

Add $1$ and $1$ .

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{{\sqrt{29}}^{2}}$

Step 6.3.2.6

Rewrite ${\sqrt{29}}^{2}$ as $29$ .

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Step 6.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{29}$ as $29^{\frac{1}{2}}$ .

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{{(29^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

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

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{29^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

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

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{29^{\frac{2}{2}}}$

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.

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{29^{\frac{2}{2}}}$

Step 6.3.2.6.4.2

Rewrite the expression.

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{29^{1}}$

Step 6.3.2.6.5

Evaluate the exponent.

$\sin (x + 0.38050637) = \frac{\sqrt{29}}{29}$

Step 7

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

$x + 0.38050637 = \sin^{-1} (\frac{\sqrt{29}}{29})$

Step 8

Simplify the right side.

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Step 8.1

Evaluate $\sin^{-1} (\frac{\sqrt{29}}{29})$ .

$x + 0.38050637 = 0.18677946$

Step 9

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

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Step 9.1

Subtract $0.38050637$ from both sides of the equation.

$x = 0.18677946 - 0.38050637$

Step 9.2

Subtract $0.38050637$ from $0.18677946$ .

$x = - 0.19372691$

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.

$x + 0.38050637 = (3.14159265) - 0.18677946$

Step 11

Solve for $x$ .

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Step 11.1

Subtract $0.18677946$ from $3.14159265$ .

$x + 0.38050637 = 2.95481319$

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 $0.38050637$ from both sides of the equation.

$x = 2.95481319 - 0.38050637$

Step 11.2.2

Subtract $0.38050637$ from $2.95481319$ .

$x = 2.57430681$

Step 12

Find the period of $\sin (x + 0.38050637)$ .

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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 $- 0.19372691$ to find the positive angle.

$- 0.19372691 + 2 π$

Step 13.2

Subtract $0.19372691$ from $2 π$ .

$6.08945839$

Step 13.3

List the new angles.

$x = 6.08945839$

Step 14

The period of the $\sin (x + 0.38050637)$ function is $2 π$ so values will repeat every $2 π$ radians in both directions.

$x = 2.57430681 + 2 π n, 6.08945839 + 2 π n$ , for any integer $n$