Trigonometry Examples

$2 \cos (h (2 x)) - \sin (h (2 x)) = 2$

Step 1

Subtract $2$ from both sides of the equation.

$2 \cos (h \cdot (2 x)) - \sin (h \cdot (2 x)) - 2 = 0$

Step 2

Simplify the left side of the equation.

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Step 2.1

Simplify terms.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite using the commutative property of multiplication.

$2 \cos (2 h x) - \sin (h \cdot (2 x)) - 2 = 0$

Step 2.1.1.2

Add parentheses.

$2 \cos (2 (h x)) - \sin (h \cdot (2 x)) - 2 = 0$

Step 2.1.1.3

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

$2 (\cos^{2} (h x) - \sin^{2} (h x)) - \sin (h \cdot (2 x)) - 2 = 0$

Step 2.1.1.4

Apply the distributive property.

$2 \cos^{2} (h x) + 2 (- \sin^{2} (h x)) - \sin (h \cdot (2 x)) - 2 = 0$

Step 2.1.1.5

Multiply $- 1$ by $2$ .

$2 \cos^{2} (h x) - 2 \sin^{2} (h x) - \sin (h \cdot (2 x)) - 2 = 0$

Step 2.1.1.6

Rewrite using the commutative property of multiplication.

$2 \cos^{2} (h x) - 2 \sin^{2} (h x) - \sin (2 h x) - 2 = 0$

Step 2.1.1.7

Add parentheses.

$2 \cos^{2} (h x) - 2 \sin^{2} (h x) - \sin (2 (h x)) - 2 = 0$

Step 2.1.1.8

Apply the sine double-angle identity.

$2 \cos^{2} (h x) - 2 \sin^{2} (h x) - (2 \sin (h x) \cos (h x)) - 2 = 0$

Step 2.1.1.9

Multiply $2$ by $- 1$ .

$2 \cos^{2} (h x) - 2 \sin^{2} (h x) - 2 \sin (h x) \cos (h x) - 2 = 0$

Step 2.1.2

Simplify with factoring out.

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Step 2.1.2.1

Move $- 2$ .

$2 \cos^{2} (h x) - 2 - 2 \sin^{2} (h x) - 2 \sin (h x) \cos (h x) = 0$

Step 2.1.2.2

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

$- 2 + 2 \cos^{2} (h x) - 2 \sin^{2} (h x) - 2 \sin (h x) \cos (h x) = 0$

Step 2.1.2.3

Factor $- 2$ out of $- 2$ .

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

Step 2.1.2.4

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

$- 2 \cdot 1 - 2 (- \cos^{2} (h x)) - 2 \sin^{2} (h x) - 2 \sin (h x) \cos (h x) = 0$

Step 2.1.2.5

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

$- 2 (1 - \cos^{2} (h x)) - 2 \sin^{2} (h x) - 2 \sin (h x) \cos (h x) = 0$

Step 2.2

Apply pythagorean identity.

$- 2 \sin^{2} (h x) - 2 \sin^{2} (h x) - 2 \sin (h x) \cos (h x) = 0$

Step 2.3

Subtract $2 \sin^{2} (h x)$ from $- 2 \sin^{2} (h x)$ .

$- 4 \sin^{2} (h x) - 2 \sin (h x) \cos (h x) = 0$

Step 3

Factor $- 4 \sin^{2} (h x) - 2 \sin (h x) \cos (h x)$ .

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Step 3.1

Factor $2 \sin (h x)$ out of $- 4 \sin^{2} (h x) - 2 \sin (h x) \cos (h x)$ .

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Step 3.1.1

Factor $2 \sin (h x)$ out of $- 4 \sin^{2} (h x)$ .

$2 \sin (h x) (- 2 \sin (h x)) - 2 \sin (h x) \cos (h x) = 0$

Step 3.1.2

Factor $2 \sin (h x)$ out of $- 2 \sin (h x) \cos (h x)$ .

$2 \sin (h x) (- 2 \sin (h x)) + 2 \sin (h x) (- 1 \cos (h x)) = 0$

Step 3.1.3

Factor $2 \sin (h x)$ out of $2 \sin (h x) (- 2 \sin (h x)) + 2 \sin (h x) (- 1 \cos (h x))$ .

$2 \sin (h x) (- 2 \sin (h x) - 1 \cos (h x)) = 0$

Step 3.2

Rewrite $- 1 \cos (h x)$ as $- \cos (h x)$ .

$2 \sin (h x) (- 2 \sin (h x) - \cos (h x)) = 0$

Step 4

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

$\sin (h x) = 0$

$- 2 \sin (h x) - \cos (h x) = 0$

Step 5

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

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Step 5.1

Set $\sin (h x)$ equal to $0$ .

$\sin (h x) = 0$

Step 5.2

Solve $\sin (h x) = 0$ for $x$ .

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Step 5.2.1

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

$h x = \arcsin (0)$

Step 5.2.2

Simplify the right side.

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Step 5.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$h x = 0$

Step 5.2.3

Divide each term in $h x = 0$ by $h$ and simplify.

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Step 5.2.3.1

Divide each term in $h x = 0$ by $h$ .

$\frac{h x}{h} = \frac{0}{h}$

Step 5.2.3.2

Simplify the left side.

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Step 5.2.3.2.1

Cancel the common factor of $h$ .

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Step 5.2.3.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{0}{h}$

Step 5.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{h}$

Step 5.2.3.3

Simplify the right side.

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Step 5.2.3.3.1

Divide $0$ by $h$ .

$x = 0$

Step 5.2.4

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.

$h x = π - 0$

Step 5.2.5

Solve for $x$ .

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Step 5.2.5.1

Simplify.

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Step 5.2.5.1.1

Multiply $- 1$ by $0$ .

$h x = π + 0$

Step 5.2.5.1.2

Add $π$ and $0$ .

$h x = π$

Step 5.2.5.2

Divide each term in $h x = π$ by $h$ and simplify.

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Step 5.2.5.2.1

Divide each term in $h x = π$ by $h$ .

$\frac{h x}{h} = \frac{π}{h}$

Step 5.2.5.2.2

Simplify the left side.

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Step 5.2.5.2.2.1

Cancel the common factor of $h$ .

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Step 5.2.5.2.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{π}{h}$

Step 5.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{π}{h}$

Step 6

Set $- 2 \sin (h x) - \cos (h x)$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $- 2 \sin (h x) - \cos (h x)$ equal to $0$ .

$- 2 \sin (h x) - \cos (h x) = 0$

Step 6.2

Solve $- 2 \sin (h x) - \cos (h x) = 0$ for $x$ .

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Step 6.2.1

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

$\frac{- 2 \sin (h x)}{\cos (h x)} + \frac{- \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 6.2.2

Separate fractions.

$\frac{- 2}{1} \cdot \frac{\sin (h x)}{\cos (h x)} + \frac{- \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 6.2.3

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

$\frac{- 2}{1} \cdot \tan (h x) + \frac{- \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 6.2.4

Divide $- 2$ by $1$ .

$- 2 \tan (h x) + \frac{- \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 6.2.5

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

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Step 6.2.5.1

Cancel the common factor.

$- 2 \tan (h x) + \frac{- \cos (h x)}{\cos (h x)} = \frac{0}{\cos (h x)}$

Step 6.2.5.2

Divide $- 1$ by $1$ .

$- 2 \tan (h x) - 1 = \frac{0}{\cos (h x)}$

Step 6.2.6

Separate fractions.

$- 2 \tan (h x) - 1 = \frac{0}{1} \cdot \frac{1}{\cos (h x)}$

Step 6.2.7

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

$- 2 \tan (h x) - 1 = \frac{0}{1} \cdot \sec (h x)$

Step 6.2.8

Divide $0$ by $1$ .

$- 2 \tan (h x) - 1 = 0 \sec (h x)$

Step 6.2.9

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

$- 2 \tan (h x) - 1 = 0$

Step 6.2.10

Add $1$ to both sides of the equation.

$- 2 \tan (h x) = 1$

Step 6.2.11

Divide each term in $- 2 \tan (h x) = 1$ by $- 2$ and simplify.

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Step 6.2.11.1

Divide each term in $- 2 \tan (h x) = 1$ by $- 2$ .

$\frac{- 2 \tan (h x)}{- 2} = \frac{1}{- 2}$

Step 6.2.11.2

Simplify the left side.

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Step 6.2.11.2.1

Cancel the common factor of $- 2$ .

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Step 6.2.11.2.1.1

Cancel the common factor.

$\frac{- 2 \tan (h x)}{- 2} = \frac{1}{- 2}$

Step 6.2.11.2.1.2

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

$\tan (h x) = \frac{1}{- 2}$

Step 6.2.11.3

Simplify the right side.

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Step 6.2.11.3.1

Move the negative in front of the fraction.

$\tan (h x) = - \frac{1}{2}$

Step 6.2.12

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

$h x = \arctan (- \frac{1}{2})$

Step 6.2.13

Simplify the right side.

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Step 6.2.13.1

Evaluate $\arctan (- \frac{1}{2})$ .

$h x = - 0.4636476$

Step 6.2.14

Divide each term in $h x = - 0.4636476$ by $h$ and simplify.

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Step 6.2.14.1

Divide each term in $h x = - 0.4636476$ by $h$ .

$\frac{h x}{h} = \frac{- 0.4636476}{h}$

Step 6.2.14.2

Simplify the left side.

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Step 6.2.14.2.1

Cancel the common factor of $h$ .

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Step 6.2.14.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{- 0.4636476}{h}$

Step 6.2.14.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 0.4636476}{h}$

Step 6.2.14.3

Simplify the right side.

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Step 6.2.14.3.1

Move the negative in front of the fraction.

$x = - \frac{0.4636476}{h}$

Step 6.2.15

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

$h x = - 0.4636476 - (3.14159265)$

Step 6.2.16

Add $2 π$ to $- 0.4636476 - (3.14159265)$ .

$h x = - 0.4636476 - (3.14159265) + 2 π$

Step 6.2.17

The resulting angle of $2.67794504$ is positive and coterminal with $- 0.4636476 - (3.14159265)$ .

$h x = 2.67794504$

Step 6.2.18

Divide each term in $h x = 2.67794504$ by $h$ and simplify.

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Step 6.2.18.1

Divide each term in $h x = 2.67794504$ by $h$ .

$\frac{h x}{h} = \frac{2.67794504}{h}$

Step 6.2.18.2

Simplify the left side.

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Step 6.2.18.2.1

Cancel the common factor of $h$ .

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Step 6.2.18.2.1.1

Cancel the common factor.

$\frac{h x}{h} = \frac{2.67794504}{h}$

Step 6.2.18.2.1.2

Divide $x$ by $1$ .

$x = \frac{2.67794504}{h}$

Step 7

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

$x = \frac{π}{h}$

$x = \frac{2.67794504}{h}$