Precalculus Examples

$\sin (x) = \frac{15}{17}$ , $\cos (2 x) = y$

Step 1

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

$x = \arcsin (\frac{15}{17})$

Step 2

Simplify the right side.

Tap for more steps...

Evaluate $\arcsin (\frac{15}{17})$ .

$x = 1.080839$

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.

$x = (3.14159265) - 1.080839$

Step 4

Solve for $x$ .

Tap for more steps...

Remove parentheses.

$x = 3.14159265 - 1.080839$

Remove parentheses.

$x = (3.14159265) - 1.080839$

Subtract $1.080839$ from $3.14159265$ .

$x = 2.06075365$

Step 5

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

Tap for more steps...

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

$\frac{2 π}{| b |}$

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

$\frac{2 π}{| 1 |}$

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

$\frac{2 π}{1}$

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.

$x = 1.080839 + 2 π n, 2.06075365 + 2 π n$ , for any integer $n$

Step 7

Reorder $1.080839$ and $2 π n$ .

$x = 2 π n + 1.080839$

$2.06075365 + 2 π n$

Step 8

Solve the equation for $n$ .

Tap for more steps...

Simplify each term.

Tap for more steps...

Apply the distributive property.

$(2 \cdot 2.06075365 + 2 (2 π n)) \cdot n + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

Multiply $2$ by $2.06075365$ .

$(4.1215073 + 2 (2 π n)) \cdot n + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

Multiply $2$ by $2$ .

$(4.1215073 + 4 (π n)) \cdot n + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

Apply the distributive property.

$4.1215073 n + 4 π n \cdot n + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

Multiply $n$ by $n$ by adding the exponents.

Tap for more steps...

Move $n$ .

$4.1215073 n + 4 π (n \cdot n) + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

Multiply $n$ by $n$ .

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 2 (2.06075365 + 2 π n) \cdot n$

$2.06075365 + 2 π n$

Simplify each term.

Tap for more steps...

Apply the distributive property.

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + (2 \cdot 2.06075365 + 2 (2 π n)) \cdot n$

$2.06075365 + 2 π n$

Multiply $2$ by $2.06075365$ .

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + (4.1215073 + 2 (2 π n)) \cdot n$

$2.06075365 + 2 π n$

Multiply $2$ by $2$ .

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + (4.1215073 + 4 (π n)) \cdot n$

$2.06075365 + 2 π n$

Apply the distributive property.

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 4.1215073 n + 4 π n \cdot n$

$2.06075365 + 2 π n$

Multiply $n$ by $n$ by adding the exponents.

Tap for more steps...

Move $n$ .

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 4.1215073 n + 4 π (n \cdot n)$

$2.06075365 + 2 π n$

Multiply $n$ by $n$ .

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 4.1215073 n + 4 π n^{2}$

$2.06075365 + 2 π n$

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 4.1215073 n + 4 π n^{2}$

$2.06075365 + 2 π n$

$4.1215073 n + 4 π n^{2} + 1.080839 = 1.080839 + 4.1215073 n + 4 π n^{2}$

$2.06075365 + 2 π n$

Move all terms containing $n$ to the left side of the equation.

Tap for more steps...

Subtract $4.1215073 n$ from both sides of the equation.

$4.1215073 n + 4 π n^{2} + 1.080839 - 4.1215073 n = 1.080839 + 4 π n^{2}$

$2.06075365 + 2 π n$

Subtract $4 π n^{2}$ from both sides of the equation.

$4.1215073 n + 4 π n^{2} + 1.080839 - 4.1215073 n - 4 π n^{2} = 1.080839$

$2.06075365 + 2 π n$

Combine the opposite terms in $4.1215073 n + 4 π n^{2} + 1.080839 - 4.1215073 n - 4 π n^{2}$ .

Tap for more steps...

Subtract $4.1215073 n$ from $4.1215073 n$ .

$0 n + 4 π n^{2} + 1.080839 - 4 π n^{2} = 1.080839$

$2.06075365 + 2 π n$

Subtract $4 π n^{2}$ from $4 π n^{2}$ .

$0 n + 0 + 1.080839 = 1.080839$

$2.06075365 + 2 π n$

Add $0 n$ and $0$ .

$0 n + 1.080839 = 1.080839$

$2.06075365 + 2 π n$

$0 n + 1.080839 = 1.080839$

$2.06075365 + 2 π n$

Multiply $0$ by $n$ .

$0 + 1.080839 = 1.080839$

$2.06075365 + 2 π n$

Add $0$ and $1.080839$ .

$1.080839 = 1.080839$

$2.06075365 + 2 π n$

$1.080839 = 1.080839$

$2.06075365 + 2 π n$

Since $1.080839 = 1.080839$ , the equation will always be true.

Always true

$2.06075365 + 2 π n$

Always true

$2.06075365 + 2 π n$

Step 9

The simplified system is the arbitrary solution of the original system of equations.

Always true

$2.06075365 + 2 (2.06075365 + 2 π n) \cdot n$