Trigonometry Examples

$2 \cos (2 x) + 7 \sin (x) = 5$

Step 1

Subtract $5$ from both sides of the equation.

$2 \cos (2 x) + 7 \sin (x) - 5 = 0$

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify each term.

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Step 2.1.1

Use the double-angle identity to transform $\cos (2 x)$ to $1 - 2 \sin^{2} (x)$ .

$2 (1 - 2 \sin^{2} (x)) + 7 \sin (x) - 5 = 0$

Step 2.1.2

Apply the distributive property.

$2 \cdot 1 + 2 (- 2 \sin^{2} (x)) + 7 \sin (x) - 5 = 0$

Step 2.1.3

Multiply $2$ by $1$ .

$2 + 2 (- 2 \sin^{2} (x)) + 7 \sin (x) - 5 = 0$

Step 2.1.4

Multiply $- 2$ by $2$ .

$2 - 4 \sin^{2} (x) + 7 \sin (x) - 5 = 0$

Step 2.2

Subtract $5$ from $2$ .

$- 3 - 4 \sin^{2} (x) + 7 \sin (x) = 0$

Step 3

Factor by grouping.

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Step 3.1

Reorder terms.

$- 4 \sin^{2} (x) + 7 \sin (x) - 3 = 0$

Step 3.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 4 \cdot - 3 = 12$ and whose sum is $b = 7$ .

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Step 3.2.1

Factor $7$ out of $7 \sin (x)$ .

$- 4 \sin^{2} (x) + 7 \sin (x) - 3 = 0$

Step 3.2.2

Rewrite $7$ as $3$ plus $4$

$- 4 \sin^{2} (x) + (3 + 4) \sin (x) - 3 = 0$

Step 3.2.3

Apply the distributive property.

$- 4 \sin^{2} (x) + 3 \sin (x) + 4 \sin (x) - 3 = 0$

Step 3.3

Factor out the greatest common factor from each group.

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Step 3.3.1

Group the first two terms and the last two terms.

$- 4 \sin^{2} (x) + 3 \sin (x) + 4 \sin (x) - 3 = 0$

Step 3.3.2

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

$\sin (x) (- 4 \sin (x) + 3) - (- 4 \sin (x) + 3) = 0$

Step 3.4

Factor the polynomial by factoring out the greatest common factor, $- 4 \sin (x) + 3$ .

$(- 4 \sin (x) + 3) (\sin (x) - 1) = 0$

Step 4

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

$- 4 \sin (x) + 3 = 0$

$\sin (x) - 1 = 0$

Step 5

Set $- 4 \sin (x) + 3$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $- 4 \sin (x) + 3$ equal to $0$ .

$- 4 \sin (x) + 3 = 0$

Step 5.2

Solve $- 4 \sin (x) + 3 = 0$ for $x$ .

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Step 5.2.1

Subtract $3$ from both sides of the equation.

$- 4 \sin (x) = - 3$

Step 5.2.2

Divide each term in $- 4 \sin (x) = - 3$ by $- 4$ and simplify.

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Step 5.2.2.1

Divide each term in $- 4 \sin (x) = - 3$ by $- 4$ .

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

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $- 4$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

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

Step 5.2.2.2.1.2

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

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

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Dividing two negative values results in a positive value.

$\sin (x) = \frac{3}{4}$

Step 5.2.3

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

$x = \arcsin (\frac{3}{4})$

Step 5.2.4

Simplify the right side.

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Step 5.2.4.1

Evaluate $\arcsin (\frac{3}{4})$ .

$x = 0.84806207$

Step 5.2.5

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.

$x = (3.14159265) - 0.84806207$

Step 5.2.6

Solve for $x$ .

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Step 5.2.6.1

Remove parentheses.

$x = 3.14159265 - 0.84806207$

Step 5.2.6.2

Remove parentheses.

$x = (3.14159265) - 0.84806207$

Step 5.2.6.3

Subtract $0.84806207$ from $3.14159265$ .

$x = 2.29353057$

Step 5.2.7

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

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Step 5.2.7.1

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

$\frac{2 π}{| b |}$

Step 5.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 5.2.7.3

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

$\frac{2 π}{1}$

Step 5.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 5.2.8

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

$x = 0.84806207 + 2 π n, 2.29353057 + 2 π n$ , for any integer $n$

Step 6

Set $\sin (x) - 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $\sin (x) - 1$ equal to $0$ .

$\sin (x) - 1 = 0$

Step 6.2

Solve $\sin (x) - 1 = 0$ for $x$ .

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Step 6.2.1

Add $1$ to both sides of the equation.

$\sin (x) = 1$

Step 6.2.2

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

$x = \arcsin (1)$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$x = \frac{π}{2}$

Step 6.2.4

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.

$x = π - \frac{π}{2}$

Step 6.2.5

Simplify $π - \frac{π}{2}$ .

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Step 6.2.5.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.2.5.2

Combine fractions.

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Step 6.2.5.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 6.2.5.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 6.2.5.3

Simplify the numerator.

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Step 6.2.5.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 6.2.5.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 6.2.6

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

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Step 6.2.6.1

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

$\frac{2 π}{| b |}$

Step 6.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 6.2.6.3

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

$\frac{2 π}{1}$

Step 6.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.2.7

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

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

Step 7

The final solution is all the values that make $(- 4 \sin (x) + 3) (\sin (x) - 1) = 0$ true.

$x = 0.84806207 + 2 π n, 2.29353057 + 2 π n, \frac{π}{2} + 2 π n$ , for any integer $n$