Trigonometry Examples

$7 \cos (9 x) - 1 = 1$

Step 1

Move all terms not containing $\cos (9 x)$ to the right side of the equation.

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Step 1.1

Add $1$ to both sides of the equation.

$7 \cos (9 x) = 1 + 1$

Step 1.2

Add $1$ and $1$ .

$7 \cos (9 x) = 2$

Step 2

Divide each term in $7 \cos (9 x) = 2$ by $7$ and simplify.

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Step 2.1

Divide each term in $7 \cos (9 x) = 2$ by $7$ .

$\frac{7 \cos (9 x)}{7} = \frac{2}{7}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $7$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{7 \cos (9 x)}{7} = \frac{2}{7}$

Step 2.2.1.2

Divide $\cos (9 x)$ by $1$ .

$\cos (9 x) = \frac{2}{7}$

Step 3

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

$9 x = \arccos (\frac{2}{7})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arccos (\frac{2}{7})$ .

$9 x = 73.3984504$

Step 5

Divide each term in $9 x = 73.3984504$ by $9$ and simplify.

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Step 5.1

Divide each term in $9 x = 73.3984504$ by $9$ .

$\frac{9 x}{9} = \frac{73.3984504}{9}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $9$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{73.3984504}{9}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{73.3984504}{9}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $73.3984504$ by $9$ .

$x = 8.15538337$

Step 6

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $360$ to find the solution in the fourth quadrant.

$9 x = 360 - 73.3984504$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $73.3984504$ from $360$ .

$9 x = 286.60154959$

Step 7.2

Divide each term in $9 x = 286.60154959$ by $9$ and simplify.

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Step 7.2.1

Divide each term in $9 x = 286.60154959$ by $9$ .

$\frac{9 x}{9} = \frac{286.60154959}{9}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $9$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{9 x}{9} = \frac{286.60154959}{9}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{286.60154959}{9}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $286.60154959$ by $9$ .

$x = 31.84461662$

Step 8

Find the period of $\cos (9 x)$ .

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Step 8.1

The period of the function can be calculated using $\frac{360}{| b |}$ .

$\frac{360}{| b |}$

Step 8.2

Replace $b$ with $9$ in the formula for period.

$\frac{360}{| 9 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $9$ is $9$ .

$\frac{360}{9}$

Step 8.4

Divide $360$ by $9$ .

$40$

Step 9

The period of the $\cos (9 x)$ function is $40$ so values will repeat every $40$ degrees in both directions.

$x = 8.15538337 + 40 n, 31.84461662 + 40 n$ , for any integer $n$