Trigonometry Examples

$\frac{1}{\cos (x)} = \frac{8}{3}$

Step 1

Find the LCD of the terms in the equation.

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Step 1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$\cos (x), 3$

Step 1.2

Since $\cos (x), 3$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 3$ then find LCM for the variable part $\cos^{1} (x)$ .

Step 1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 1.4

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 1.5

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 1.6

The LCM of $1, 3$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 1.7

The factor for $\cos^{1} (x)$ is $\cos (x)$ itself.

$\cos^{1} (x) = \cos (x)$

$\cos (x)$ occurs $1$ time.

Step 1.8

The LCM of $\cos^{1} (x)$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$\cos (x)$

Step 1.9

The LCM for $\cos (x), 3$ is the numeric part $3$ multiplied by the variable part.

$3 \cos (x)$

Step 2

Multiply each term in $\frac{1}{\cos (x)} = \frac{8}{3}$ by $3 \cos (x)$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{1}{\cos (x)} = \frac{8}{3}$ by $3 \cos (x)$ .

$\frac{1}{\cos (x)} (3 \cos (x)) = \frac{8}{3} (3 \cos (x))$

Step 2.2

Simplify the left side.

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Step 2.2.1

Rewrite using the commutative property of multiplication.

$3 \frac{1}{\cos (x)} \cos (x) = \frac{8}{3} (3 \cos (x))$

Step 2.2.2

Combine $3$ and $\frac{1}{\cos (x)}$ .

$\frac{3}{\cos (x)} \cos (x) = \frac{8}{3} (3 \cos (x))$

Step 2.2.3

Cancel the common factor of $\cos (x)$ .

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Step 2.2.3.1

Cancel the common factor.

$\frac{3}{\cos (x)} \cos (x) = \frac{8}{3} (3 \cos (x))$

Step 2.2.3.2

Rewrite the expression.

$3 = \frac{8}{3} (3 \cos (x))$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $3$ .

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Step 2.3.1.1

Factor $3$ out of $3 \cos (x)$ .

$3 = \frac{8}{3} (3 \cos (x))$

Step 2.3.1.2

Cancel the common factor.

$3 = \frac{8}{3} (3 \cos (x))$

Step 2.3.1.3

Rewrite the expression.

$3 = 8 \cos (x)$

Step 3

Solve the equation.

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Step 3.1

Rewrite the equation as $8 \cos (x) = 3$ .

$8 \cos (x) = 3$

Step 3.2

Divide each term in $8 \cos (x) = 3$ by $8$ and simplify.

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Step 3.2.1

Divide each term in $8 \cos (x) = 3$ by $8$ .

$\frac{8 \cos (x)}{8} = \frac{3}{8}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $8$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{8 \cos (x)}{8} = \frac{3}{8}$

Step 3.2.2.1.2

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

$\cos (x) = \frac{3}{8}$

Step 4

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

$x = \arccos (\frac{3}{8})$

Step 5

Simplify the right side.

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Step 5.1

Evaluate $\arccos (\frac{3}{8})$ .

$x = 1.18639955$

Step 6

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.

$x = 2 (3.14159265) - 1.18639955$

Step 7

Simplify $2 (3.14159265) - 1.18639955$ .

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Step 7.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.18639955$

Step 7.2

Subtract $1.18639955$ from $6.2831853$ .

$x = 5.09678575$

Step 8

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

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Step 8.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 8.2

Replace $b$ with $1$ in the formula for period.

$\frac{2 π}{| 1 |}$

Step 8.3

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

$\frac{2 π}{1}$

Step 8.4

Divide $2 π$ by $1$ .

$2 π$

Step 9

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

$x = 1.18639955 + 2 π n, 5.09678575 + 2 π n$ , for any integer $n$