Trigonometry Examples

$(\sin (x) + 0.2) (\cos (2 x) - 1) = 0$

Step 1

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

$\sin (x) + 0.2 = 0$

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

Step 2

Set $\sin (x) + 0.2$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $\sin (x) + 0.2$ equal to $0$ .

$\sin (x) + 0.2 = 0$

Step 2.2

Solve $\sin (x) + 0.2 = 0$ for $x$ .

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Step 2.2.1

Subtract $0.2$ from both sides of the equation.

$\sin (x) = - 0.2$

Step 2.2.2

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

$x = \arcsin (- 0.2)$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Evaluate $\arcsin (- 0.2)$ .

$x = - 0.20135792$

Step 2.2.4

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.

$x = 2 (3.14159265) + 0.20135792 + 3.14159265$

Step 2.2.5

Simplify the expression to find the second solution.

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Step 2.2.5.1

Subtract $2 π$ from $2 (3.14159265) + 0.20135792 + 3.14159265$ .

$x = 2 (3.14159265) + 0.20135792 + 3.14159265 - 2 π$

Step 2.2.5.2

The resulting angle of $3.34295057$ is positive, less than $2 π$ , and coterminal with $2 (3.14159265) + 0.20135792 + 3.14159265$ .

$x = 3.34295057$

Step 2.2.6

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

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Step 2.2.6.1

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

$\frac{2 π}{| b |}$

Step 2.2.6.2

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

$\frac{2 π}{| 1 |}$

Step 2.2.6.3

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

$\frac{2 π}{1}$

Step 2.2.6.4

Divide $2 π$ by $1$ .

$2 π$

Step 2.2.7

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

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Step 2.2.7.1

Add $2 π$ to $- 0.20135792$ to find the positive angle.

$- 0.20135792 + 2 π$

Step 2.2.7.2

Subtract $0.20135792$ from $2 π$ .

$6.08182738$

Step 2.2.7.3

List the new angles.

$x = 6.08182738$

Step 2.2.8

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

$x = 3.34295057 + 2 π n, 6.08182738 + 2 π n$ , for any integer $n$

Step 3

Set $\cos (2 x) - 1$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $\cos (2 x) - 1$ equal to $0$ .

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

Step 3.2

Solve $\cos (2 x) - 1 = 0$ for $x$ .

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Step 3.2.1

Add $1$ to both sides of the equation.

$\cos (2 x) = 1$

Step 3.2.2

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

$2 x = \arccos (1)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

The exact value of $\arccos (1)$ is $0$ .

$2 x = 0$

Step 3.2.4

Divide each term in $2 x = 0$ by $2$ and simplify.

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Step 3.2.4.1

Divide each term in $2 x = 0$ by $2$ .

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.4.2

Simplify the left side.

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Step 3.2.4.2.1

Cancel the common factor of $2$ .

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Step 3.2.4.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0}{2}$

Step 3.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{2}$

Step 3.2.4.3

Simplify the right side.

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Step 3.2.4.3.1

Divide $0$ by $2$ .

$x = 0$

Step 3.2.5

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$2 x = 2 π - 0$

Step 3.2.6

Solve for $x$ .

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Step 3.2.6.1

Simplify.

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Step 3.2.6.1.1

Multiply $- 1$ by $0$ .

$2 x = 2 π + 0$

Step 3.2.6.1.2

Add $2 π$ and $0$ .

$2 x = 2 π$

Step 3.2.6.2

Divide each term in $2 x = 2 π$ by $2$ and simplify.

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Step 3.2.6.2.1

Divide each term in $2 x = 2 π$ by $2$ .

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 3.2.6.2.2

Simplify the left side.

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Step 3.2.6.2.2.1

Cancel the common factor of $2$ .

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Step 3.2.6.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{2 π}{2}$

Step 3.2.6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2 π}{2}$

Step 3.2.6.2.3

Simplify the right side.

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Step 3.2.6.2.3.1

Cancel the common factor of $2$ .

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Step 3.2.6.2.3.1.1

Cancel the common factor.

$x = \frac{2 π}{2}$

Step 3.2.6.2.3.1.2

Divide $π$ by $1$ .

$x = π$

Step 3.2.7

Find the period of $\cos (2 x)$ .

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Step 3.2.7.1

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

$\frac{2 π}{| b |}$

Step 3.2.7.2

Replace $b$ with $2$ in the formula for period.

$\frac{2 π}{| 2 |}$

Step 3.2.7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$\frac{2 π}{2}$

Step 3.2.7.4

Cancel the common factor of $2$ .

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Step 3.2.7.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 3.2.7.4.2

Divide $π$ by $1$ .

$π$

Step 3.2.8

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

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

Step 4

The final solution is all the values that make $(\sin (x) + 0.2) (\cos (2 x) - 1) = 0$ true.

$x = 3.34295057 + 2 π n, 6.08182738 + 2 π n, π n, π + π n$ , for any integer $n$

Step 5

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

$x = 3.34295057 + 2 π n, 6.08182738 + 2 π n, π n$ , for any integer $n$