Precalculus Examples

$\sin (x) = \frac{12}{13}$ , $\cos (x) = - \frac{5}{13}$

Step 1

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

$x = \arcsin (\frac{12}{13})$

$\cos (x) = - \frac{5}{13}$

Step 2

Simplify the right side.

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Step 2.1

Evaluate $\arcsin (\frac{12}{13})$ .

$x = 1.1760052$

$\cos (x) = - \frac{5}{13}$

$x = 1.1760052$

$\cos (x) = - \frac{5}{13}$

Step 3

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

$x = (3.14159265) - 1.1760052$

$\cos (x) = - \frac{5}{13}$

Step 4

Solve for $x$ .

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Step 4.1

Remove parentheses.

$x = 3.14159265 - 1.1760052$

$\cos (x) = - \frac{5}{13}$

Step 4.2

Remove parentheses.

$x = (3.14159265) - 1.1760052$

$\cos (x) = - \frac{5}{13}$

Step 4.3

Subtract $1.1760052$ from $3.14159265$ .

$x = 1.96558744$

$\cos (x) = - \frac{5}{13}$

$x = 1.96558744$

$\cos (x) = - \frac{5}{13}$

Step 5

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

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Step 5.1

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

$\frac{2 π}{| b |}$

Step 5.2

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

$\frac{2 π}{| 1 |}$

Step 5.3

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

$\frac{2 π}{1}$

Step 5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6

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

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

$\cos (x) = - \frac{5}{13}$

Step 7

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

$x = \arccos (- \frac{5}{13})$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

Step 8

Simplify the right side.

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Step 8.1

Evaluate $\arccos (- \frac{5}{13})$ .

$x = 1.96558744$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

$x = 1.96558744$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

Step 9

The cosine function is negative in the second and third quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the third quadrant.

$x = 2 (3.14159265) - 1.96558744$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

Step 10

Solve for $x$ .

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Step 10.1

Remove parentheses.

$x = 2 (3.14159265) - 1.96558744$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

Step 10.2

Simplify $2 (3.14159265) - 1.96558744$ .

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Step 10.2.1

Multiply $2$ by $3.14159265$ .

$x = 6.2831853 - 1.96558744$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

Step 10.2.2

Subtract $1.96558744$ from $6.2831853$ .

$x = 4.31759786$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

$x = 4.31759786$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

$x = 4.31759786$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

Step 11

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

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Step 11.1

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

$\frac{2 π}{| b |}$

Step 11.2

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

$\frac{2 π}{| 1 |}$

Step 11.3

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

$\frac{2 π}{1}$

Step 11.4

Divide $2 π$ by $1$ .

$2 π$

Step 12

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

$x = 1.96558744 + 2 π n, 4.31759786 + 2 π n$

$x = 1.1760052 + 2 π n, 1.96558744 + 2 π n$

Step 13

Since the roots of an equation are the points where the solution is $0$ , set each root as a factor of the equation that equals $0$ .

$(x - (1.1760052 + 2 π n, 1.96558744 + 2 π n)) (x - (1.96558744 + 2 π n, 4.31759786 + 2 π n)) = 0$