Trigonometry Examples

$5 \cos (x + 3) = 1$

Step 1

Divide each term in $5 \cos (x + 3) = 1$ by $5$ and simplify.

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Step 1.1

Divide each term in $5 \cos (x + 3) = 1$ by $5$ .

$\frac{5 \cos (x + 3)}{5} = \frac{1}{5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{5 \cos (x + 3)}{5} = \frac{1}{5}$

Step 1.2.1.2

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

$\cos (x + 3) = \frac{1}{5}$

Step 2

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

$x + 3 = \arccos (\frac{1}{5})$

Step 3

Simplify the right side.

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Step 3.1

Evaluate $\arccos (\frac{1}{5})$ .

$x + 3 = 1.3694384$

Step 4

Move all terms not containing $x$ to the right side of the equation.

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Step 4.1

Subtract $3$ from both sides of the equation.

$x = 1.3694384 - 3$

Step 4.2

Subtract $3$ from $1.3694384$ .

$x = - 1.63056159$

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.

$x + 3 = 2 (3.14159265) - 1.3694384$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $2 (3.14159265) - 1.3694384$ .

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Step 6.1.1

Multiply $2$ by $3.14159265$ .

$x + 3 = 6.2831853 - 1.3694384$

Step 6.1.2

Subtract $1.3694384$ from $6.2831853$ .

$x + 3 = 4.9137469$

Step 6.2

Move all terms not containing $x$ to the right side of the equation.

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Step 6.2.1

Subtract $3$ from both sides of the equation.

$x = 4.9137469 - 3$

Step 6.2.2

Subtract $3$ from $4.9137469$ .

$x = 1.9137469$

Step 7

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

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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 $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 7.3

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{2 π}{1}$

Step 7.4

Divide $2 π$ by $1$ .

$2 π$

Step 8

Add $2 π$ to every negative angle to get positive angles.

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Step 8.1

Add $2 π$ to $- 1.63056159$ to find the positive angle.

$- 1.63056159 + 2 π$

Step 8.2

Subtract $1.63056159$ from $2 π$ .

$4.65262371$

Step 8.3

List the new angles.

$x = 4.65262371$

Step 9

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

$x = 1.9137469 + 2 π n, 4.65262371 + 2 π n$ , for any integer $n$