Trigonometry Examples

$\cos (3 x - 45) = \frac{1}{2}$

Step 1

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

$3 x - 45 = \arccos (\frac{1}{2})$

Step 2

Simplify the right side.

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Step 2.1

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

$3 x - 45 = \frac{π}{3}$

Step 3

Add $45$ to both sides of the equation.

$3 x = \frac{π}{3} + 45$

Step 4

Divide each term in $3 x = \frac{π}{3} + 45$ by $3$ and simplify.

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Step 4.1

Divide each term in $3 x = \frac{π}{3} + 45$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{π}{3}}{3} + \frac{45}{3}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $3$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{π}{3}}{3} + \frac{45}{3}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{3}}{3} + \frac{45}{3}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{3} \cdot \frac{1}{3} + \frac{45}{3}$

Step 4.3.1.2

Multiply $\frac{π}{3} \cdot \frac{1}{3}$ .

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Step 4.3.1.2.1

Multiply $\frac{π}{3}$ by $\frac{1}{3}$ .

$x = \frac{π}{3 \cdot 3} + \frac{45}{3}$

Step 4.3.1.2.2

Multiply $3$ by $3$ .

$x = \frac{π}{9} + \frac{45}{3}$

Step 4.3.1.3

Divide $45$ by $3$ .

$x = \frac{π}{9} + 15$

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.

$3 x - 45 = 2 π - \frac{π}{3}$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 π - \frac{π}{3}$ .

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Step 6.1.1

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

$3 x - 45 = 2 π \cdot \frac{3}{3} - \frac{π}{3}$

Step 6.1.2

Combine fractions.

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Step 6.1.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$3 x - 45 = \frac{2 π \cdot 3}{3} - \frac{π}{3}$

Step 6.1.2.2

Combine the numerators over the common denominator.

$3 x - 45 = \frac{2 π \cdot 3 - π}{3}$

Step 6.1.3

Simplify the numerator.

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Step 6.1.3.1

Multiply $3$ by $2$ .

$3 x - 45 = \frac{6 π - π}{3}$

Step 6.1.3.2

Subtract $π$ from $6 π$ .

$3 x - 45 = \frac{5 π}{3}$

Step 6.2

Add $45$ to both sides of the equation.

$3 x = \frac{5 π}{3} + 45$

Step 6.3

Divide each term in $3 x = \frac{5 π}{3} + 45$ by $3$ and simplify.

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Step 6.3.1

Divide each term in $3 x = \frac{5 π}{3} + 45$ by $3$ .

$\frac{3 x}{3} = \frac{\frac{5 π}{3}}{3} + \frac{45}{3}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $3$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{\frac{5 π}{3}}{3} + \frac{45}{3}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{5 π}{3}}{3} + \frac{45}{3}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Simplify each term.

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Step 6.3.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{5 π}{3} \cdot \frac{1}{3} + \frac{45}{3}$

Step 6.3.3.1.2

Multiply $\frac{5 π}{3} \cdot \frac{1}{3}$ .

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Step 6.3.3.1.2.1

Multiply $\frac{5 π}{3}$ by $\frac{1}{3}$ .

$x = \frac{5 π}{3 \cdot 3} + \frac{45}{3}$

Step 6.3.3.1.2.2

Multiply $3$ by $3$ .

$x = \frac{5 π}{9} + \frac{45}{3}$

Step 6.3.3.1.3

Divide $45$ by $3$ .

$x = \frac{5 π}{9} + 15$

Step 7

Find the period of $\cos (3 x - 45)$ .

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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 $3$ 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 $\cos (3 x - 45)$ function is $\frac{2 π}{3}$ so values will repeat every $\frac{2 π}{3}$ radians in both directions.

$x = \frac{π}{9} + 15 + \frac{2 π n}{3}, \frac{5 π}{9} + 15 + \frac{2 π n}{3}$ , for any integer $n$