Trigonometry Examples

$4 \cos (θ) + 1 = 2 \cos (θ)$

Step 1

Move all terms containing $\cos (θ)$ to the left side of the equation.

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Step 1.1

Subtract $2 \cos (θ)$ from both sides of the equation.

$4 \cos (θ) + 1 - 2 \cos (θ) = 0$

Step 1.2

Subtract $2 \cos (θ)$ from $4 \cos (θ)$ .

$2 \cos (θ) + 1 = 0$

Step 2

Subtract $1$ from both sides of the equation.

$2 \cos (θ) = - 1$

Step 3

Divide each term in $2 \cos (θ) = - 1$ by $2$ and simplify.

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Step 3.1

Divide each term in $2 \cos (θ) = - 1$ by $2$ .

$\frac{2 \cos (θ)}{2} = \frac{- 1}{2}$

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.

$\frac{2 \cos (θ)}{2} = \frac{- 1}{2}$

Step 3.2.1.2

Divide $\cos (θ)$ by $1$ .

$\cos (θ) = \frac{- 1}{2}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative in front of the fraction.

$\cos (θ) = - \frac{1}{2}$

Step 4

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

$θ = \arccos (- \frac{1}{2})$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$θ = \frac{2 π}{3}$

Step 6

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.

$θ = 2 π - \frac{2 π}{3}$

Step 7

Simplify $2 π - \frac{2 π}{3}$ .

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Step 7.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$θ = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$θ = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 7.2.2

Combine the numerators over the common denominator.

$θ = \frac{2 π \cdot 3 - 2 π}{3}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Multiply $3$ by $2$ .

$θ = \frac{6 π - 2 π}{3}$

Step 7.3.2

Subtract $2 π$ from $6 π$ .

$θ = \frac{4 π}{3}$

Step 8

Find the period of $\cos (θ)$ .

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Step 8.1

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

$\frac{2 π}{| b |}$

Step 8.2

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

$\frac{2 π}{| 1 |}$

Step 8.3

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

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

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

$θ = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$