Trigonometry Examples

$2 \cos (3 x) = 1.5$

Step 1

Divide each term in $2 \cos (3 x) = 1.5$ by $2$ and simplify.

Tap for more steps...

Step 1.1

Divide each term in $2 \cos (3 x) = 1.5$ by $2$ .

$\frac{2 \cos (3 x)}{2} = \frac{1.5}{2}$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.1.1

Cancel the common factor.

$\frac{2 \cos (3 x)}{2} = \frac{1.5}{2}$

Step 1.2.1.2

Divide $\cos (3 x)$ by $1$ .

$\cos (3 x) = \frac{1.5}{2}$

Step 1.3

Simplify the right side.

Tap for more steps...

Step 1.3.1

Divide $1.5$ by $2$ .

$\cos (3 x) = 0.75$

Step 2

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

$3 x = \arccos (0.75)$

Step 3

Simplify the right side.

Tap for more steps...

Step 3.1

Evaluate $\arccos (0.75)$ .

$3 x = 0.72273424$

Step 4

Divide each term in $3 x = 0.72273424$ by $3$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $3 x = 0.72273424$ by $3$ .

$\frac{3 x}{3} = \frac{0.72273424}{3}$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0.72273424}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{0.72273424}{3}$

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Divide $0.72273424$ by $3$ .

$x = 0.24091141$

Step 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.

$3 x = 2 (3.14159265) - 0.72273424$

Step 6

Solve for $x$ .

Tap for more steps...

Step 6.1

Simplify.

Tap for more steps...

Step 6.1.1

Multiply $2$ by $3.14159265$ .

$3 x = 6.2831853 - 0.72273424$

Step 6.1.2

Subtract $0.72273424$ from $6.2831853$ .

$3 x = 5.56045105$

Step 6.2

Divide each term in $3 x = 5.56045105$ by $3$ and simplify.

Tap for more steps...

Step 6.2.1

Divide each term in $3 x = 5.56045105$ by $3$ .

$\frac{3 x}{3} = \frac{5.56045105}{3}$

Step 6.2.2

Simplify the left side.

Tap for more steps...

Step 6.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 6.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{5.56045105}{3}$

Step 6.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{5.56045105}{3}$

Step 6.2.3

Simplify the right side.

Tap for more steps...

Step 6.2.3.1

Divide $5.56045105$ by $3$ .

$x = 1.85348368$

Step 7

Find the period of $\cos (3 x)$ .

Tap for more steps...

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 $3$ in the formula for period.

$\frac{2 π}{| 3 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $3$ is $3$ .

$\frac{2 π}{3}$

Step 8

The period of the $\cos (3 x)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = 0.24091141 + \frac{2 π n}{3}, 1.85348368 + \frac{2 π n}{3}$ , for any integer $n$