Trigonometry Examples

$3 \sin^{2} (x) + 2 \sin (x) = 6 \cos (x) + 9 \sin (x) \cos (x)$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $6 \cos (x)$ from both sides of the equation.

$3 \sin^{2} (x) + 2 \sin (x) - 6 \cos (x) = 9 \sin (x) \cos (x)$

Step 1.2

Subtract $9 \sin (x) \cos (x)$ from both sides of the equation.

$3 \sin^{2} (x) + 2 \sin (x) - 6 \cos (x) - 9 \sin (x) \cos (x) = 0$

Step 2

Replace the $3 \sin^{2} (x)$ with $3 (1 - \cos^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$3 (1 - \cos^{2} (x)) + 2 \sin (x) - 6 \cos (x) - 9 \sin (x) \cos (x) = 0$

Step 3

Simplify each term.

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Step 3.1

Apply the distributive property.

$3 \cdot 1 + 3 (- \cos^{2} (x)) + 2 \sin (x) - 6 \cos (x) - 9 \sin (x) \cos (x) = 0$

Step 3.2

Multiply $3$ by $1$ .

$3 + 3 (- \cos^{2} (x)) + 2 \sin (x) - 6 \cos (x) - 9 \sin (x) \cos (x) = 0$

Step 3.3

Multiply $- 1$ by $3$ .

$3 - 3 \cos^{2} (x) + 2 \sin (x) - 6 \cos (x) - 9 \sin (x) \cos (x) = 0$

Step 4

Reorder the polynomial.

$- 3 \cos^{2} (x) - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) + 3 = 0$

Step 5

Simplify the left side.

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Step 5.1

Simplify $- 3 \cos^{2} (x) - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) + 3$ .

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Step 5.1.1

Simplify with factoring out.

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Step 5.1.1.1

Move $3$ .

$- 3 \cos^{2} (x) + 3 - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) = 0$

Step 5.1.1.2

Reorder $- 3 \cos^{2} (x)$ and $3$ .

$3 - 3 \cos^{2} (x) - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) = 0$

Step 5.1.1.3

Factor $3$ out of $3$ .

$3 (1) - 3 \cos^{2} (x) - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) = 0$

Step 5.1.1.4

Factor $3$ out of $- 3 \cos^{2} (x)$ .

$3 (1) + 3 (- \cos^{2} (x)) - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) = 0$

Step 5.1.1.5

Factor $3$ out of $3 (1) + 3 (- \cos^{2} (x))$ .

$3 (1 - \cos^{2} (x)) - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) = 0$

Step 5.1.2

Apply pythagorean identity.

$3 \sin^{2} (x) - 6 \cos (x) - 9 \sin (x) \cos (x) + 2 \sin (x) = 0$

Step 6

Factor the left side of the equation.

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Step 6.1

Add parentheses.

$3 \sin^{2} (x) - 6 \cos (x) - 9 (\sin (x) \cos (x)) + 2 \sin (x) = 0$

Step 6.2

Let $u = \cos (x)$ . Substitute $u$ for all occurrences of $\cos (x)$ .

$3 \sin^{2} (x) - 6 u - 9 u \sin (x) + 2 \sin (x) = 0$

Step 6.3

Reorder terms.

$- 9 u \sin (x) - 6 u + 3 \sin^{2} (x) + 2 \sin (x) = 0$

Step 6.4

Factor out the greatest common factor from each group.

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Step 6.4.1

Group the first two terms and the last two terms.

$(- 9 u \sin (x) - 6 u) + 3 \sin^{2} (x) + 2 \sin (x) = 0$

Step 6.4.2

Factor out the greatest common factor (GCF) from each group.

$3 u (- 3 \sin (x) - 2) - \sin (x) (- 3 \sin (x) - 2) = 0$

Step 6.5

Factor the polynomial by factoring out the greatest common factor, $- 3 \sin (x) - 2$ .

$(- 3 \sin (x) - 2) (3 u - \sin (x)) = 0$

Step 6.6

Factor $- 1$ out of $- 3 \sin (x) - 2$ .

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Step 6.6.1

Factor $- 1$ out of $- 3 \sin (x)$ .

$(- (3 \sin (x)) - 2) (3 u - \sin (x)) = 0$

Step 6.6.2

Rewrite $- 2$ as $- 1 (2)$ .

$(- (3 \sin (x)) - 1 \cdot 2) (3 u - \sin (x)) = 0$

Step 6.6.3

Factor $- 1$ out of $- (3 \sin (x)) - 1 (2)$ .

$- (3 \sin (x) + 2) (3 u - \sin (x)) = 0$

Step 6.7

Replace all occurrences of $u$ with $\cos (x)$ .

$- (3 \sin (x) + 2) (3 \cos (x) - \sin (x)) = 0$

Step 7

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$3 \sin (x) + 2 = 0$

$3 \cos (x) - \sin (x) = 0$

Step 8

Set $3 \sin (x) + 2$ equal to $0$ and solve for $x$ .

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Step 8.1

Set $3 \sin (x) + 2$ equal to $0$ .

$3 \sin (x) + 2 = 0$

Step 8.2

Solve $3 \sin (x) + 2 = 0$ for $x$ .

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Step 8.2.1

Subtract $2$ from both sides of the equation.

$3 \sin (x) = - 2$

Step 8.2.2

Divide each term in $3 \sin (x) = - 2$ by $3$ and simplify.

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Step 8.2.2.1

Divide each term in $3 \sin (x) = - 2$ by $3$ .

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

Step 8.2.2.2

Simplify the left side.

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Step 8.2.2.2.1

Cancel the common factor of $3$ .

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Step 8.2.2.2.1.1

Cancel the common factor.

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

Step 8.2.2.2.1.2

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

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

Step 8.2.2.3

Simplify the right side.

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Step 8.2.2.3.1

Move the negative in front of the fraction.

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

Step 8.2.3

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

$x = \arcsin (- \frac{2}{3})$

Step 8.2.4

Simplify the right side.

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Step 8.2.4.1

Evaluate $\arcsin (- \frac{2}{3})$ .

$x = - 0.72972765$

Step 8.2.5

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) + 0.72972765 + 3.14159265$

Step 8.2.6

Simplify the expression to find the second solution.

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Step 8.2.6.1

Subtract $2 π$ from $2 (3.14159265) + 0.72972765 + 3.14159265$ .

$x = 2 (3.14159265) + 0.72972765 + 3.14159265 - 2 π$

Step 8.2.6.2

The resulting angle of $3.8713203$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.72972765 + 3.14159265$ .

$x = 3.8713203$

Step 8.2.7

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

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Step 8.2.7.1

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

$\frac{2 π}{| b |}$

Step 8.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 8.2.7.3

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

$\frac{2 π}{1}$

Step 8.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8.2.8

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

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Step 8.2.8.1

Add $2 π$ to $- 0.72972765$ to find the positive angle.

$- 0.72972765 + 2 π$

Step 8.2.8.2

Subtract $0.72972765$ from $2 π$ .

$5.55345765$

Step 8.2.8.3

List the new angles.

$x = 5.55345765$

Step 8.2.9

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

$x = 3.8713203 + 2 π n, 5.55345765 + 2 π n$ , for any integer $n$

Step 9

Set $3 \cos (x) - \sin (x)$ equal to $0$ and solve for $x$ .

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Step 9.1

Set $3 \cos (x) - \sin (x)$ equal to $0$ .

$3 \cos (x) - \sin (x) = 0$

Step 9.2

Solve $3 \cos (x) - \sin (x) = 0$ for $x$ .

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Step 9.2.1

Divide each term in the equation by $\cos (x)$ .

$\frac{3 \cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 9.2.2

Cancel the common factor of $\cos (x)$ .

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Step 9.2.2.1

Cancel the common factor.

$\frac{3 \cos (x)}{\cos (x)} + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 9.2.2.2

Divide $3$ by $1$ .

$3 + \frac{- \sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 9.2.3

Separate fractions.

$3 + \frac{- 1}{1} \cdot \frac{\sin (x)}{\cos (x)} = \frac{0}{\cos (x)}$

Step 9.2.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$3 + \frac{- 1}{1} \cdot \tan (x) = \frac{0}{\cos (x)}$

Step 9.2.5

Divide $- 1$ by $1$ .

$3 - \tan (x) = \frac{0}{\cos (x)}$

Step 9.2.6

Separate fractions.

$3 - \tan (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 9.2.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$3 - \tan (x) = \frac{0}{1} \cdot \sec (x)$

Step 9.2.8

Divide $0$ by $1$ .

$3 - \tan (x) = 0 \sec (x)$

Step 9.2.9

Multiply $0$ by $\sec (x)$ .

$3 - \tan (x) = 0$

Step 9.2.10

Subtract $3$ from both sides of the equation.

$- \tan (x) = - 3$

Step 9.2.11

Divide each term in $- \tan (x) = - 3$ by $- 1$ and simplify.

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Step 9.2.11.1

Divide each term in $- \tan (x) = - 3$ by $- 1$ .

$\frac{- \tan (x)}{- 1} = \frac{- 3}{- 1}$

Step 9.2.11.2

Simplify the left side.

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Step 9.2.11.2.1

Dividing two negative values results in a positive value.

$\frac{\tan (x)}{1} = \frac{- 3}{- 1}$

Step 9.2.11.2.2

Divide $\tan (x)$ by $1$ .

$\tan (x) = \frac{- 3}{- 1}$

Step 9.2.11.3

Simplify the right side.

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Step 9.2.11.3.1

Divide $- 3$ by $- 1$ .

$\tan (x) = 3$

Step 9.2.12

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

$x = \arctan (3)$

Step 9.2.13

Simplify the right side.

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Step 9.2.13.1

Evaluate $\arctan (3)$ .

$x = 1.24904577$

Step 9.2.14

The tangent function is positive in the first and third quadrants. To find the second solution, add the reference angle from $π$ to find the solution in the fourth quadrant.

$x = (3.14159265) + 1.24904577$

Step 9.2.15

Solve for $x$ .

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Step 9.2.15.1

Remove parentheses.

$x = 3.14159265 + 1.24904577$

Step 9.2.15.2

Remove parentheses.

$x = (3.14159265) + 1.24904577$

Step 9.2.15.3

Add $3.14159265$ and $1.24904577$ .

$x = 4.39063842$

Step 9.2.16

Find the period of $\tan (x)$ .

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Step 9.2.16.1

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

$\frac{π}{| b |}$

Step 9.2.16.2

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

$\frac{π}{| 1 |}$

Step 9.2.16.3

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

$\frac{π}{1}$

Step 9.2.16.4

Divide $π$ by $1$ .

$π$

Step 9.2.17

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

$x = 1.24904577 + π n, 4.39063842 + π n$ , for any integer $n$

Step 10

The final solution is all the values that make $- (3 \sin (x) + 2) (3 \cos (x) - \sin (x)) = 0$ true.

$x = 3.8713203 + 2 π n, 5.55345765 + 2 π n, 1.24904577 + π n, 4.39063842 + π n$ , for any integer $n$

Step 11

Consolidate $1.24904577 + π n$ and $4.39063842 + π n$ to $1.24904577 + π n$ .

$x = 3.8713203 + 2 π n, 5.55345765 + 2 π n, 1.24904577 + π n$ , for any integer $n$