Trigonometry Examples

$5 \sin (3 x) + 6 \cos (3 x) = 1$

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 $θ$ .

$\tan^{-1} (\frac{6}{5})$

Step 3

Evaluate $\tan^{-1} (\frac{6}{5})$ .

$θ = 0.87605805$

Step 4

Solve to find the value of $R$ .

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Step 4.1

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

$R = \sqrt{25 + {(6)}^{2}}$

Step 4.2

Raise $6$ to the power of $2$ .

$R = \sqrt{25 + 36}$

Step 4.3

Add $25$ and $36$ .

$R = \sqrt{61}$

Step 5

Substitute the known values into the equation.

$(\sqrt{61}) \sin (3 x + 0.87605805) = 1$

Step 6

Divide each term in $\sqrt{61} \sin (3 x + 0.87605805) = 1$ by $\sqrt{61}$ and simplify.

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Step 6.1

Divide each term in $\sqrt{61} \sin (3 x + 0.87605805) = 1$ by $\sqrt{61}$ .

$\frac{\sqrt{61} \sin (3 x + 0.87605805)}{\sqrt{61}} = \frac{1}{\sqrt{61}}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\sqrt{61}$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{\sqrt{61} \sin (3 x + 0.87605805)}{\sqrt{61}} = \frac{1}{\sqrt{61}}$

Step 6.2.1.2

Divide $\sin (3 x + 0.87605805)$ by $1$ .

$\sin (3 x + 0.87605805) = \frac{1}{\sqrt{61}}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Multiply $\frac{1}{\sqrt{61}}$ by $\frac{\sqrt{61}}{\sqrt{61}}$ .

$\sin (3 x + 0.87605805) = \frac{1}{\sqrt{61}} \cdot \frac{\sqrt{61}}{\sqrt{61}}$

Step 6.3.2

Combine and simplify the denominator.

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Step 6.3.2.1

Multiply $\frac{1}{\sqrt{61}}$ by $\frac{\sqrt{61}}{\sqrt{61}}$ .

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{\sqrt{61} \sqrt{61}}$

Step 6.3.2.2

Raise $\sqrt{61}$ to the power of $1$ .

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{{\sqrt{61}}^{1} \sqrt{61}}$

Step 6.3.2.3

Raise $\sqrt{61}$ to the power of $1$ .

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{{\sqrt{61}}^{1} {\sqrt{61}}^{1}}$

Step 6.3.2.4

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

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{{\sqrt{61}}^{1 + 1}}$

Step 6.3.2.5

Add $1$ and $1$ .

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{{\sqrt{61}}^{2}}$

Step 6.3.2.6

Rewrite ${\sqrt{61}}^{2}$ as $61$ .

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Step 6.3.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{61}$ as $61^{\frac{1}{2}}$ .

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{{(61^{\frac{1}{2}})}^{2}}$

Step 6.3.2.6.2

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

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{61^{\frac{1}{2} \cdot 2}}$

Step 6.3.2.6.3

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

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{61^{\frac{2}{2}}}$

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.

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{61^{\frac{2}{2}}}$

Step 6.3.2.6.4.2

Rewrite the expression.

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{61^{1}}$

Step 6.3.2.6.5

Evaluate the exponent.

$\sin (3 x + 0.87605805) = \frac{\sqrt{61}}{61}$

Step 7

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

$3 x + 0.87605805 = \sin^{-1} (\frac{\sqrt{61}}{61})$

Step 8

Simplify the right side.

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Step 8.1

Evaluate $\sin^{-1} (\frac{\sqrt{61}}{61})$ .

$3 x + 0.87605805 = 0.12838931$

Step 9

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

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Step 9.1

Subtract $0.87605805$ from both sides of the equation.

$3 x = 0.12838931 - 0.87605805$

Step 9.2

Subtract $0.87605805$ from $0.12838931$ .

$3 x = - 0.74766873$

Step 10

Divide each term in $3 x = - 0.74766873$ by $3$ and simplify.

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Step 10.1

Divide each term in $3 x = - 0.74766873$ by $3$ .

$\frac{3 x}{3} = \frac{- 0.74766873}{3}$

Step 10.2

Simplify the left side.

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Step 10.2.1

Cancel the common factor of $3$ .

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Step 10.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 0.74766873}{3}$

Step 10.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 0.74766873}{3}$

Step 10.3

Simplify the right side.

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Step 10.3.1

Divide $- 0.74766873$ by $3$ .

$x = - 0.24922291$

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.

$3 x + 0.87605805 = (3.14159265) - 0.12838931$

Step 12

Solve for $x$ .

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Step 12.1

Subtract $0.12838931$ from $3.14159265$ .

$3 x + 0.87605805 = 3.01320333$

Step 12.2

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

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Step 12.2.1

Subtract $0.87605805$ from both sides of the equation.

$3 x = 3.01320333 - 0.87605805$

Step 12.2.2

Subtract $0.87605805$ from $3.01320333$ .

$3 x = 2.13714528$

Step 12.3

Divide each term in $3 x = 2.13714528$ by $3$ and simplify.

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Step 12.3.1

Divide each term in $3 x = 2.13714528$ by $3$ .

$\frac{3 x}{3} = \frac{2.13714528}{3}$

Step 12.3.2

Simplify the left side.

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Step 12.3.2.1

Cancel the common factor of $3$ .

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Step 12.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{2.13714528}{3}$

Step 12.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.13714528}{3}$

Step 12.3.3

Simplify the right side.

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Step 12.3.3.1

Divide $2.13714528$ by $3$ .

$x = 0.71238176$

Step 13

Find the period of $\sin (3 x + 0.87605805)$ .

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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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 13.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 14

Add $\frac{2 π}{3}$ to every negative angle to get positive angles.

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Step 14.1

Add $\frac{2 π}{3}$ to $- 0.24922291$ to find the positive angle.

$- 0.24922291 + \frac{2 π}{3}$

Step 14.2

Subtract $0.24922291$ from $\frac{2 π}{3}$ .

$1.84517219$

Step 14.3

List the new angles.

$x = 1.84517219$

Step 15

The period of the $\sin (3 x + 0.87605805)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = 0.71238176 + \frac{2 π n}{3}, 1.84517219 + \frac{2 π n}{3}$ , for any integer $n$