Trigonometry Examples

$2 \cos^{2} (x) - \sin (x) - 1 = 0$ , $[0, 2 π)$

Step 1

Replace the $2 \cos^{2} (x)$ with $2 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$2 (1 - \sin^{2} (x)) - \sin (x) - 1 = 0$

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

$2 \cdot 1 + 2 (- \sin^{2} (x)) - \sin (x) - 1 = 0$

Step 2.2

Multiply $2$ by $1$ .

$2 + 2 (- \sin^{2} (x)) - \sin (x) - 1 = 0$

Step 2.3

Multiply $- 1$ by $2$ .

$2 - 2 \sin^{2} (x) - \sin (x) - 1 = 0$

Step 3

Subtract $1$ from $2$ .

$- 2 \sin^{2} (x) - \sin (x) + 1 = 0$

Step 4

Substitute $u$ for $\sin (x)$ .

$- 2 {(u)}^{2} - (u) + 1 = 0$

Step 5

Factor the left side of the equation.

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Step 5.1

Factor $- 1$ out of $- 2 u^{2} - u + 1$ .

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Step 5.1.1

Factor $- 1$ out of $- 2 u^{2}$ .

$- (2 u^{2}) - u + 1 = 0$

Step 5.1.2

Factor $- 1$ out of $- u$ .

$- (2 u^{2}) - u + 1 = 0$

Step 5.1.3

Rewrite $1$ as $- 1 (- 1)$ .

$- (2 u^{2}) - u - 1 \cdot - 1 = 0$

Step 5.1.4

Factor $- 1$ out of $- (2 u^{2}) - u$ .

$- (2 u^{2} + u) - 1 \cdot - 1 = 0$

Step 5.1.5

Factor $- 1$ out of $- (2 u^{2} + u) - 1 (- 1)$ .

$- (2 u^{2} + u - 1) = 0$

Step 5.2

Factor.

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Step 5.2.1

Factor by grouping.

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Step 5.2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 1 = - 2$ and whose sum is $b = 1$ .

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Step 5.2.1.1.1

Multiply by $1$ .

$- (2 u^{2} + 1 u - 1) = 0$

Step 5.2.1.1.2

Rewrite $1$ as $- 1$ plus $2$

$- (2 u^{2} + (- 1 + 2) u - 1) = 0$

Step 5.2.1.1.3

Apply the distributive property.

$- (2 u^{2} - 1 u + 2 u - 1) = 0$

Step 5.2.1.2

Factor out the greatest common factor from each group.

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Step 5.2.1.2.1

Group the first two terms and the last two terms.

$- (2 u^{2} - 1 u + 2 u - 1) = 0$

Step 5.2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$- (u (2 u - 1) + 1 (2 u - 1)) = 0$

Step 5.2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 u - 1$ .

$- ((2 u - 1) (u + 1)) = 0$

Step 5.2.2

Remove unnecessary parentheses.

$- (2 u - 1) (u + 1) = 0$

Step 6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$2 u - 1 = 0$

$u + 1 = 0$

Step 7

Set $2 u - 1$ equal to $0$ and solve for $u$ .

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Step 7.1

Set $2 u - 1$ equal to $0$ .

$2 u - 1 = 0$

Step 7.2

Solve $2 u - 1 = 0$ for $u$ .

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Step 7.2.1

Add $1$ to both sides of the equation.

$2 u = 1$

Step 7.2.2

Divide each term in $2 u = 1$ by $2$ and simplify.

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Step 7.2.2.1

Divide each term in $2 u = 1$ by $2$ .

$\frac{2 u}{2} = \frac{1}{2}$

Step 7.2.2.2

Simplify the left side.

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Step 7.2.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.2.1.1

Cancel the common factor.

$\frac{2 u}{2} = \frac{1}{2}$

Step 7.2.2.2.1.2

Divide $u$ by $1$ .

$u = \frac{1}{2}$

Step 8

Set $u + 1$ equal to $0$ and solve for $u$ .

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Step 8.1

Set $u + 1$ equal to $0$ .

$u + 1 = 0$

Step 8.2

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 9

The final solution is all the values that make $- (2 u - 1) (u + 1) = 0$ true.

$u = \frac{1}{2}, - 1$

Step 10

Substitute $\sin (x)$ for $u$ .

$\sin (x) = \frac{1}{2}, - 1$

Step 11

Set up each of the solutions to solve for $x$ .

$\sin (x) = \frac{1}{2}$

$\sin (x) = - 1$

Step 12

Solve for $x$ in $\sin (x) = \frac{1}{2}$ .

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Step 12.1

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

$x = \arcsin (\frac{1}{2})$

Step 12.2

Simplify the right side.

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Step 12.2.1

The exact value of $\arcsin (\frac{1}{2})$ is $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 12.3

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.

$x = π - \frac{π}{6}$

Step 12.4

Simplify $π - \frac{π}{6}$ .

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Step 12.4.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$x = π \cdot \frac{6}{6} - \frac{π}{6}$

Step 12.4.2

Combine fractions.

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Step 12.4.2.1

Combine $π$ and $\frac{6}{6}$ .

$x = \frac{π \cdot 6}{6} - \frac{π}{6}$

Step 12.4.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Step 12.4.3

Simplify the numerator.

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Step 12.4.3.1

Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Step 12.4.3.2

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

Step 12.5

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

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Step 12.5.1

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

$\frac{2 π}{| b |}$

Step 12.5.2

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

$\frac{2 π}{| 1 |}$

Step 12.5.3

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

$\frac{2 π}{1}$

Step 12.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 12.6

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

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n$ , for any integer $n$

Step 13

Solve for $x$ in $\sin (x) = - 1$ .

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Step 13.1

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

$x = \arcsin (- 1)$

Step 13.2

Simplify the right side.

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Step 13.2.1

The exact value of $\arcsin (- 1)$ is $- \frac{π}{2}$ .

$x = - \frac{π}{2}$

Step 13.3

The sine function is negative in the third and fourth quadrants. To find the second solution, subtract the solution from $2 π$ , to find a reference angle. Next, add this reference angle to $π$ to find the solution in the third quadrant.

$x = 2 π + \frac{π}{2} + π$

Step 13.4

Simplify the expression to find the second solution.

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Step 13.4.1

Subtract $2 π$ from $2 π + \frac{π}{2} + π$ .

$x = 2 π + \frac{π}{2} + π - 2 π$

Step 13.4.2

The resulting angle of $\frac{3 π}{2}$ is positive, less than $2 π$ , and coterminal with $2 π + \frac{π}{2} + π$ .

$x = \frac{3 π}{2}$

Step 13.5

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

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Step 13.5.1

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

$\frac{2 π}{| b |}$

Step 13.5.2

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

$\frac{2 π}{| 1 |}$

Step 13.5.3

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

$\frac{2 π}{1}$

Step 13.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 13.6

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

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Step 13.6.1

Add $2 π$ to $- \frac{π}{2}$ to find the positive angle.

$- \frac{π}{2} + 2 π$

Step 13.6.2

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 13.6.3

Combine fractions.

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Step 13.6.3.1

Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 13.6.3.2

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Step 13.6.4

Simplify the numerator.

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Step 13.6.4.1

Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Step 13.6.4.2

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

Step 13.6.5

List the new angles.

$x = \frac{3 π}{2}$

Step 13.7

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

$x = \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 14

List all of the solutions.

$x = \frac{π}{6} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{3 π}{2} + 2 π n, \frac{3 π}{2} + 2 π n$ , for any integer $n$

Step 15

Consolidate the answers.

$x = \frac{π}{6} + \frac{2 π n}{3}$ , for any integer $n$

Step 16

Find the values of $n$ that produce a value within the interval $[0, 2 π)$ .

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Step 16.1

Plug in $0$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 16.1.1

Plug in $0$ for $n$ .

$\frac{π}{6} + \frac{2 π (0)}{3}$

Step 16.1.2

Simplify.

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Step 16.1.2.1

Simplify each term.

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Step 16.1.2.1.1

Cancel the common factor of $0$ and $3$ .

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Step 16.1.2.1.1.1

Factor $3$ out of $2 π (0)$ .

$\frac{π}{6} + \frac{3 (2 π \cdot (0))}{3}$

Step 16.1.2.1.1.2

Cancel the common factors.

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Step 16.1.2.1.1.2.1

Factor $3$ out of $3$ .

$\frac{π}{6} + \frac{3 (2 π \cdot (0))}{3 (1)}$

Step 16.1.2.1.1.2.2

Cancel the common factor.

$\frac{π}{6} + \frac{3 (2 π \cdot (0))}{3 \cdot 1}$

Step 16.1.2.1.1.2.3

Rewrite the expression.

$\frac{π}{6} + \frac{2 π \cdot (0)}{1}$

Step 16.1.2.1.1.2.4

Divide $2 π \cdot (0)$ by $1$ .

$\frac{π}{6} + 2 π \cdot (0)$

Step 16.1.2.1.2

Multiply $2 π (0)$ .

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Step 16.1.2.1.2.1

Multiply $0$ by $2$ .

$\frac{π}{6} + 0 π$

Step 16.1.2.1.2.2

Multiply $0$ by $π$ .

$\frac{π}{6} + 0$

Step 16.1.2.2

Add $\frac{π}{6}$ and $0$ .

$\frac{π}{6}$

Step 16.1.3

The interval $[0, 2 π)$ contains $\frac{π}{6}$ .

$x = \frac{π}{6}$

Step 16.2

Plug in $1$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 16.2.1

Plug in $1$ for $n$ .

$\frac{π}{6} + \frac{2 π (1)}{3}$

Step 16.2.2

Simplify.

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Step 16.2.2.1

Multiply $2$ by $1$ .

$\frac{π}{6} + \frac{2 π}{3}$

Step 16.2.2.2

To write $\frac{2 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{π}{6} + \frac{2 π}{3} \cdot \frac{2}{2}$

Step 16.2.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.2.2.3.1

Multiply $\frac{2 π}{3}$ by $\frac{2}{2}$ .

$\frac{π}{6} + \frac{2 π \cdot 2}{3 \cdot 2}$

Step 16.2.2.3.2

Multiply $3$ by $2$ .

$\frac{π}{6} + \frac{2 π \cdot 2}{6}$

Step 16.2.2.4

Combine the numerators over the common denominator.

$\frac{π + 2 π \cdot 2}{6}$

Step 16.2.2.5

Simplify the numerator.

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Step 16.2.2.5.1

Multiply $2$ by $2$ .

$\frac{π + 4 π}{6}$

Step 16.2.2.5.2

Add $π$ and $4 π$ .

$\frac{5 π}{6}$

Step 16.2.3

The interval $[0, 2 π)$ contains $\frac{5 π}{6}$ .

$x = \frac{π}{6}, \frac{5 π}{6}$

Step 16.3

Plug in $2$ for $n$ and simplify to see if the solution is contained in $[0, 2 π)$ .

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Step 16.3.1

Plug in $2$ for $n$ .

$\frac{π}{6} + \frac{2 π (2)}{3}$

Step 16.3.2

Simplify.

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Step 16.3.2.1

Multiply $2$ by $2$ .

$\frac{π}{6} + \frac{4 π}{3}$

Step 16.3.2.2

To write $\frac{4 π}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{π}{6} + \frac{4 π}{3} \cdot \frac{2}{2}$

Step 16.3.2.3

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 16.3.2.3.1

Multiply $\frac{4 π}{3}$ by $\frac{2}{2}$ .

$\frac{π}{6} + \frac{4 π \cdot 2}{3 \cdot 2}$

Step 16.3.2.3.2

Multiply $3$ by $2$ .

$\frac{π}{6} + \frac{4 π \cdot 2}{6}$

Step 16.3.2.4

Combine the numerators over the common denominator.

$\frac{π + 4 π \cdot 2}{6}$

Step 16.3.2.5

Simplify the numerator.

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Step 16.3.2.5.1

Multiply $2$ by $4$ .

$\frac{π + 8 π}{6}$

Step 16.3.2.5.2

Add $π$ and $8 π$ .

$\frac{9 π}{6}$

Step 16.3.2.6

Cancel the common factor of $9$ and $6$ .

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Step 16.3.2.6.1

Factor $3$ out of $9 π$ .

$\frac{3 (3 π)}{6}$

Step 16.3.2.6.2

Cancel the common factors.

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Step 16.3.2.6.2.1

Factor $3$ out of $6$ .

$\frac{3 (3 π)}{3 (2)}$

Step 16.3.2.6.2.2

Cancel the common factor.

$\frac{3 (3 π)}{3 \cdot 2}$

Step 16.3.2.6.2.3

Rewrite the expression.

$\frac{3 π}{2}$

Step 16.3.3

The interval $[0, 2 π)$ contains $\frac{3 π}{2}$ .

$x = \frac{π}{6}, \frac{5 π}{6}, \frac{3 π}{2}$