Trigonometry Examples

$\cos (2 x) = \frac{4}{5}$

Step 1

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

$2 x = \arccos (\frac{4}{5})$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arccos (\frac{4}{5})$ .

$2 x = 0.6435011$

Step 3

Divide each term in $2 x = 0.6435011$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 x = 0.6435011$ by $2$ .

$\frac{2 x}{2} = \frac{0.6435011}{2}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $2$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{0.6435011}{2}$

Step 3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.6435011}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Divide $0.6435011$ by $2$ .

$x = 0.32175055$

Step 4

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.

$2 x = 2 (3.14159265) - 0.6435011$

Step 5

Solve for $x$ .

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $2$ by $3.14159265$ .

$2 x = 6.2831853 - 0.6435011$

Step 5.1.2

Subtract $0.6435011$ from $6.2831853$ .

$2 x = 5.63968419$

Step 5.2

Divide each term in $2 x = 5.63968419$ by $2$ and simplify.

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Step 5.2.1

Divide each term in $2 x = 5.63968419$ by $2$ .

$\frac{2 x}{2} = \frac{5.63968419}{2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{5.63968419}{2}$

Step 5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5.63968419}{2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Divide $5.63968419$ by $2$ .

$x = 2.81984209$

Step 6

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

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Step 6.1

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

$\frac{2 π}{| b |}$

Step 6.2

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

$\frac{2 π}{| 2 |}$

Step 6.3

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

$\frac{2 π}{2}$

Step 6.4

Cancel the common factor of $2$ .

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Step 6.4.1

Cancel the common factor.

$\frac{2 π}{2}$

Step 6.4.2

Divide $π$ by $1$ .

$π$

Step 7

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

$x = 0.32175055 + π n, 2.81984209 + π n$ , for any integer $n$