Precalculus Examples

$\cos (x) + 1 = \sin (x)$

Step 1

Subtract $1$ from both sides of the equation.

$\cos (x) = \sin (x) - 1$

Step 2

Square both sides of the equation.

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

Step 3

Simplify ${(\sin (x) - 1)}^{2}$ .

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Step 3.1

Rewrite ${(\sin (x) - 1)}^{2}$ as $(\sin (x) - 1) (\sin (x) - 1)$ .

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

Step 3.2

Expand $(\sin (x) - 1) (\sin (x) - 1)$ using the FOIL Method.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Apply the distributive property.

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

Step 3.2.3

Apply the distributive property.

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

Step 3.3

Simplify and combine like terms.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Multiply $\sin (x) \sin (x)$ .

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Step 3.3.1.1.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 3.3.1.1.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 3.3.1.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.3.1.1.4

Add $1$ and $1$ .

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

Step 3.3.1.2

Move $- 1$ to the left of $\sin (x)$ .

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

Step 3.3.1.3

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

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

Step 3.3.1.4

Rewrite $- 1 \sin (x)$ as $- \sin (x)$ .

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

Step 3.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 3.3.2

Subtract $\sin (x)$ from $- \sin (x)$ .

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

Step 4

Move all the expressions to the left side of the equation.

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Step 4.1

Subtract $\sin^{2} (x)$ from both sides of the equation.

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

Step 4.2

Add $2 \sin (x)$ to both sides of the equation.

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

Step 4.3

Subtract $1$ from both sides of the equation.

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

Step 5

Simplify $\cos^{2} (x) - \sin^{2} (x) + 2 \sin (x) - 1$ .

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Step 5.1

Move $- 1$ .

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

Step 5.2

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 5.3

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

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

Step 5.4

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 5.5

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 5.6

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 5.7

Apply pythagorean identity.

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

Step 5.8

Subtract $\sin^{2} (x)$ from $- \sin^{2} (x)$ .

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

Step 6

Solve for $x$ .

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Step 6.1

Factor the left side of the equation.

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Step 6.1.1

Let $u = \sin (x)$ . Substitute $u$ for all occurrences of $\sin (x)$ .

$- 2 u^{2} + 2 u = 0$

Step 6.1.2

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

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Step 6.1.2.1

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

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

Step 6.1.2.2

Factor $2 u$ out of $2 u$ .

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

Step 6.1.2.3

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

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

Step 6.1.3

Replace all occurrences of $u$ with $\sin (x)$ .

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

Step 6.2

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

$\sin (x) = 0$

$- \sin (x) + 1 = 0$

Step 6.3

Set $\sin (x)$ equal to $0$ and solve for $x$ .

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Step 6.3.1

Set $\sin (x)$ equal to $0$ .

$\sin (x) = 0$

Step 6.3.2

Solve $\sin (x) = 0$ for $x$ .

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Step 6.3.2.1

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

$x = \arcsin (0)$

Step 6.3.2.2

Simplify the right side.

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Step 6.3.2.2.1

The exact value of $\arcsin (0)$ is $0$ .

$x = 0$

Step 6.3.2.3

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.

$x = π - 0$

Step 6.3.2.4

Subtract $0$ from $π$ .

$x = π$

Step 6.3.2.5

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

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Step 6.3.2.5.1

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

$\frac{2 π}{| b |}$

Step 6.3.2.5.2

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

$\frac{2 π}{| 1 |}$

Step 6.3.2.5.3

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

$\frac{2 π}{1}$

Step 6.3.2.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.3.2.6

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

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

Step 6.4

Set $- \sin (x) + 1$ equal to $0$ and solve for $x$ .

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Step 6.4.1

Set $- \sin (x) + 1$ equal to $0$ .

$- \sin (x) + 1 = 0$

Step 6.4.2

Solve $- \sin (x) + 1 = 0$ for $x$ .

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Step 6.4.2.1

Subtract $1$ from both sides of the equation.

$- \sin (x) = - 1$

Step 6.4.2.2

Divide each term in $- \sin (x) = - 1$ by $- 1$ and simplify.

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Step 6.4.2.2.1

Divide each term in $- \sin (x) = - 1$ by $- 1$ .

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

Step 6.4.2.2.2

Simplify the left side.

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Step 6.4.2.2.2.1

Dividing two negative values results in a positive value.

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

Step 6.4.2.2.2.2

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

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

Step 6.4.2.2.3

Simplify the right side.

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Step 6.4.2.2.3.1

Divide $- 1$ by $- 1$ .

$\sin (x) = 1$

Step 6.4.2.3

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

$x = \arcsin (1)$

Step 6.4.2.4

Simplify the right side.

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Step 6.4.2.4.1

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

$x = \frac{π}{2}$

Step 6.4.2.5

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.

$x = π - \frac{π}{2}$

Step 6.4.2.6

Simplify $π - \frac{π}{2}$ .

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Step 6.4.2.6.1

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

$x = π \cdot \frac{2}{2} - \frac{π}{2}$

Step 6.4.2.6.2

Combine fractions.

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Step 6.4.2.6.2.1

Combine $π$ and $\frac{2}{2}$ .

$x = \frac{π \cdot 2}{2} - \frac{π}{2}$

Step 6.4.2.6.2.2

Combine the numerators over the common denominator.

$x = \frac{π \cdot 2 - π}{2}$

Step 6.4.2.6.3

Simplify the numerator.

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Step 6.4.2.6.3.1

Move $2$ to the left of $π$ .

$x = \frac{2 \cdot π - π}{2}$

Step 6.4.2.6.3.2

Subtract $π$ from $2 π$ .

$x = \frac{π}{2}$

Step 6.4.2.7

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

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Step 6.4.2.7.1

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

$\frac{2 π}{| b |}$

Step 6.4.2.7.2

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

$\frac{2 π}{| 1 |}$

Step 6.4.2.7.3

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

$\frac{2 π}{1}$

Step 6.4.2.7.4

Divide $2 π$ by $1$ .

$2 π$

Step 6.4.2.8

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

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

Step 6.5

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

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

Step 7

Consolidate $2 π n$ and $π + 2 π n$ to $π n$ .

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

Step 8

Exclude the solutions that do not make $\cos (x) + 1 = \sin (x)$ true.

$x = 1.57079632, 3.14159265, 7.85398163, 9.42477796, 14.13716694, 15.70796326, 20.42035224, 21.99114857, 26.70353755, 32.98672286, 39.26990816, 45.55309347$ , for any integer $n$