Trigonometry Examples

$7 \cos (5 x) + 9 = 12$

Step 1

Move all terms not containing $\cos (5 x)$ to the right side of the equation.

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Step 1.1

Subtract $9$ from both sides of the equation.

$7 \cos (5 x) = 12 - 9$

Step 1.2

Subtract $9$ from $12$ .

$7 \cos (5 x) = 3$

Step 2

Divide each term in $7 \cos (5 x) = 3$ by $7$ and simplify.

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Step 2.1

Divide each term in $7 \cos (5 x) = 3$ by $7$ .

$\frac{7 \cos (5 x)}{7} = \frac{3}{7}$

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.

$\frac{7 \cos (5 x)}{7} = \frac{3}{7}$

Step 2.2.1.2

Divide $\cos (5 x)$ by $1$ .

$\cos (5 x) = \frac{3}{7}$

Step 3

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

$5 x = \arccos (\frac{3}{7})$

Step 4

Simplify the right side.

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Step 4.1

Evaluate $\arccos (\frac{3}{7})$ .

$5 x = 64.62306647$

Step 5

Divide each term in $5 x = 64.62306647$ by $5$ and simplify.

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Step 5.1

Divide each term in $5 x = 64.62306647$ by $5$ .

$\frac{5 x}{5} = \frac{64.62306647}{5}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $5$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{64.62306647}{5}$

Step 5.2.1.2

Divide $x$ by $1$ .

$x = \frac{64.62306647}{5}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $64.62306647$ by $5$ .

$x = 12.92461329$

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.

$5 x = 360 - 64.62306647$

Step 7

Solve for $x$ .

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Step 7.1

Subtract $64.62306647$ from $360$ .

$5 x = 295.37693352$

Step 7.2

Divide each term in $5 x = 295.37693352$ by $5$ and simplify.

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Step 7.2.1

Divide each term in $5 x = 295.37693352$ by $5$ .

$\frac{5 x}{5} = \frac{295.37693352}{5}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $5$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{295.37693352}{5}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{295.37693352}{5}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $295.37693352$ by $5$ .

$x = 59.0753867$

Step 8

Find the period of $\cos (5 x)$ .

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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 $5$ in the formula for period.

$\frac{360}{| 5 |}$

Step 8.3

The absolute value is the distance between a number and zero. The distance between $0$ and $5$ is $5$ .

$\frac{360}{5}$

Step 8.4

Divide $360$ by $5$ .

$72$

Step 9

The period of the $\cos (5 x)$ function is $72$ so values will repeat every $72$ degrees in both directions.

$x = 12.92461329 + 72 n, 59.0753867 + 72 n$ , for any integer $n$