Trigonometry Examples

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

Step 1

Move all terms not containing $\cos (x)$ to the right side of the equation.

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Step 1.1

Subtract $1$ from both sides of the equation.

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

Step 1.2

Subtract $1$ from $- 1$ .

$2 \cos (x) = - 2$

Step 2

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

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Step 2.1

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

$\frac{2 \cos (x)}{2} = \frac{- 2}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{2 \cos (x)}{2} = \frac{- 2}{2}$

Step 2.2.1.2

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

$\cos (x) = \frac{- 2}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 2$ by $2$ .

$\cos (x) = - 1$

Step 3

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

$x = \arccos (- 1)$

Step 4

Simplify the right side.

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Step 4.1

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

$x = π$

Step 5

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

$x = 2 π - π$

Step 6

Subtract $π$ from $2 π$ .

$x = π$

Step 7

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

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Step 7.1

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

$\frac{2 π}{| b |}$

Step 7.2

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

$\frac{2 π}{| 1 |}$

Step 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 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

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

$x = π + 2 π n$ , for any integer $n$