Trigonometry Examples

$- \frac{\sqrt{3}}{2} + 3 \cos (x) = 4 \cos (x)$

Step 1

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

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Step 1.1

Subtract $4 \cos (x)$ from both sides of the equation.

$- \frac{\sqrt{3}}{2} + 3 \cos (x) - 4 \cos (x) = 0$

Step 1.2

Subtract $4 \cos (x)$ from $3 \cos (x)$ .

$- \frac{\sqrt{3}}{2} - \cos (x) = 0$

Step 2

Add $\frac{\sqrt{3}}{2}$ to both sides of the equation.

$- \cos (x) = \frac{\sqrt{3}}{2}$

Step 3

Divide each term in $- \cos (x) = \frac{\sqrt{3}}{2}$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- \cos (x) = \frac{\sqrt{3}}{2}$ by $- 1$ .

$\frac{- \cos (x)}{- 1} = \frac{\frac{\sqrt{3}}{2}}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (x)}{1} = \frac{\frac{\sqrt{3}}{2}}{- 1}$

Step 3.2.2

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

$\cos (x) = \frac{\frac{\sqrt{3}}{2}}{- 1}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Move the negative one from the denominator of $\frac{\frac{\sqrt{3}}{2}}{- 1}$ .

$\cos (x) = - 1 \cdot \frac{\sqrt{3}}{2}$

Step 3.3.2

Rewrite $- 1 \cdot \frac{\sqrt{3}}{2}$ as $- \frac{\sqrt{3}}{2}$ .

$\cos (x) = - \frac{\sqrt{3}}{2}$

Step 4

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

$x = \arccos (- \frac{\sqrt{3}}{2})$

Step 5

Simplify the right side.

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Step 5.1

The exact value of $\arccos (- \frac{\sqrt{3}}{2})$ is $\frac{5 π}{6}$ .

$x = \frac{5 π}{6}$

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.

$x = 2 π - \frac{5 π}{6}$

Step 7

Simplify $2 π - \frac{5 π}{6}$ .

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Step 7.1

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = 2 π \cdot \frac{6}{6} - \frac{5 π}{6}$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $2 π$ and $\frac{6}{6}$ .

$x = \frac{2 π \cdot 6}{6} - \frac{5 π}{6}$

Step 7.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 6 - 5 π}{6}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Multiply $6$ by $2$ .

$x = \frac{12 π - 5 π}{6}$

Step 7.3.2

Subtract $5 π$ from $12 π$ .

$x = \frac{7 π}{6}$

Step 8

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

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

$x = \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$