Trigonometry Examples

$\sin (x) + \sqrt{3} \cos (x) = - 1$

Step 1

Subtract $\sin (x)$ from both sides of the equation.

$\sqrt{3} \cos (x) = - 1 - \sin (x)$

Step 2

Square both sides of the equation.

${(\sqrt{3} \cos (x))}^{2} = {(- 1 - \sin (x))}^{2}$

Step 3

Simplify ${(\sqrt{3} \cos (x))}^{2}$ .

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Apply the product rule to $\sqrt{3} \cos (x)$ .

${\sqrt{3}}^{2} \cos^{2} (x) = {(- 1 - \sin (x))}^{2}$

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

${(3^{\frac{1}{2}})}^{2} \cos^{2} (x) = {(- 1 - \sin (x))}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$3^{\frac{1}{2} \cdot 2} \cos^{2} (x) = {(- 1 - \sin (x))}^{2}$

Combine $\frac{1}{2}$ and $2$ .

$3^{\frac{2}{2}} \cos^{2} (x) = {(- 1 - \sin (x))}^{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$3^{\frac{2}{2}} \cos^{2} (x) = {(- 1 - \sin (x))}^{2}$

Rewrite the expression.

$3^{1} \cos^{2} (x) = {(- 1 - \sin (x))}^{2}$

Evaluate the exponent.

$3 \cos^{2} (x) = {(- 1 - \sin (x))}^{2}$

Step 4

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

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

$3 \cos^{2} (x) = (- 1 - \sin (x)) (- 1 - \sin (x))$

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

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Apply the distributive property.

$3 \cos^{2} (x) = - 1 (- 1 - \sin (x)) - \sin (x) (- 1 - \sin (x))$

Apply the distributive property.

$3 \cos^{2} (x) = - 1 \cdot - 1 - 1 (- \sin (x)) - \sin (x) (- 1 - \sin (x))$

Apply the distributive property.

$3 \cos^{2} (x) = - 1 \cdot - 1 - 1 (- \sin (x)) - \sin (x) \cdot - 1 - \sin (x) (- \sin (x))$

Simplify and combine like terms.

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Simplify each term.

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Multiply $- 1$ by $- 1$ .

$3 \cos^{2} (x) = 1 - 1 (- \sin (x)) - \sin (x) \cdot - 1 - \sin (x) (- \sin (x))$

Multiply $- 1 (- \sin (x))$ .

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Multiply $- 1$ by $- 1$ .

$3 \cos^{2} (x) = 1 + 1 \sin (x) - \sin (x) \cdot - 1 - \sin (x) (- \sin (x))$

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

$3 \cos^{2} (x) = 1 + \sin (x) - \sin (x) \cdot - 1 - \sin (x) (- \sin (x))$

Multiply $- \sin (x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$3 \cos^{2} (x) = 1 + \sin (x) + 1 \sin (x) - \sin (x) (- \sin (x))$

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

$3 \cos^{2} (x) = 1 + \sin (x) + \sin (x) - \sin (x) (- \sin (x))$

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

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Multiply $- 1$ by $- 1$ .

$3 \cos^{2} (x) = 1 + \sin (x) + \sin (x) + 1 \sin (x) \sin (x)$

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

$3 \cos^{2} (x) = 1 + \sin (x) + \sin (x) + \sin (x) \sin (x)$

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

$3 \cos^{2} (x) = 1 + \sin (x) + \sin (x) + \sin^{1} (x) \sin (x)$

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

$3 \cos^{2} (x) = 1 + \sin (x) + \sin (x) + \sin^{1} (x) \sin^{1} (x)$

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

$3 \cos^{2} (x) = 1 + \sin (x) + \sin (x) + {\sin (x)}^{1 + 1}$

Add $1$ and $1$ .

$3 \cos^{2} (x) = 1 + \sin (x) + \sin (x) + \sin^{2} (x)$

Add $\sin (x)$ and $\sin (x)$ .

$3 \cos^{2} (x) = 1 + 2 \sin (x) + \sin^{2} (x)$

Step 5

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

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Subtract $1$ from both sides of the equation.

$3 \cos^{2} (x) - 1 = 2 \sin (x) + \sin^{2} (x)$

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

$3 \cos^{2} (x) - 1 - 2 \sin (x) = \sin^{2} (x)$

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

$3 \cos^{2} (x) - 1 - 2 \sin (x) - \sin^{2} (x) = 0$

Step 6

Replace the $3 \cos^{2} (x)$ with $3 (1 - \sin^{2} (x))$ based on the $\sin^{2} (x) + \cos^{2} (x) = 1$ identity.

$3 (1 - \sin^{2} (x)) - 1 - 2 \sin (x) - \sin^{2} (x) = 0$

Step 7

Simplify each term.

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Apply the distributive property.

$3 \cdot 1 + 3 (- \sin^{2} (x)) - 1 - 2 \sin (x) - \sin^{2} (x) = 0$

Multiply $3$ by $1$ .

$3 + 3 (- \sin^{2} (x)) - 1 - 2 \sin (x) - \sin^{2} (x) = 0$

Multiply $- 1$ by $3$ .

$3 - 3 \sin^{2} (x) - 1 - 2 \sin (x) - \sin^{2} (x) = 0$

Step 8

Simplify by adding terms.

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Subtract $1$ from $3$ .

$- 3 \sin^{2} (x) + 2 - 2 \sin (x) - \sin^{2} (x) = 0$

Subtract $\sin^{2} (x)$ from $- 3 \sin^{2} (x)$ .

$- 4 \sin^{2} (x) + 2 - 2 \sin (x) = 0$

Step 9

Reorder the polynomial.

$- 4 \sin^{2} (x) - 2 \sin (x) + 2 = 0$

Step 10

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

$- 4 {(u)}^{2} - 2 u + 2 = 0$

Step 11

Factor the left side of the equation.

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Factor $- 2$ out of $- 4 u^{2} - 2 u + 2$ .

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Factor $- 2$ out of $- 4 u^{2}$ .

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

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

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

Factor $- 2$ out of $2$ .

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

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

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

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

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

Factor.

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Factor by grouping.

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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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Multiply by $1$ .

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

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

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

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

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

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

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

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

Remove unnecessary parentheses.

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

Step 12

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 13

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

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

$2 u - 1 = 0$

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

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Add $1$ to both sides of the equation.

$2 u = 1$

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

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

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

Simplify the left side.

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Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $u$ by $1$ .

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

Step 14

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

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

$u + 1 = 0$

Subtract $1$ from both sides of the equation.

$u = - 1$

Step 15

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

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

Step 16

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

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

Step 17

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

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

$\sin (x) = - 1$

Step 18

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

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

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

Simplify the right side.

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

$x = \frac{π}{6}$

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.

$x = π - \frac{π}{6}$

Simplify $π - \frac{π}{6}$ .

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

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

Combine fractions.

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Combine $π$ and $\frac{6}{6}$ .

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

Combine the numerators over the common denominator.

$x = \frac{π \cdot 6 - π}{6}$

Simplify the numerator.

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Move $6$ to the left of $π$ .

$x = \frac{6 \cdot π - π}{6}$

Subtract $π$ from $6 π$ .

$x = \frac{5 π}{6}$

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

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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 $1$ .

$2 π$

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 19

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

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

$x = \arcsin (- 1)$

Simplify the right side.

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

$x = - \frac{π}{2}$

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.

$x = 2 π + \frac{π}{2} + π$

Simplify the expression to find the second solution.

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

$x = 2 π + \frac{π}{2} + π - 2 π$

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

$x = \frac{3 π}{2}$

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

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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 $1$ .

$2 π$

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

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

$- \frac{π}{2} + 2 π$

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

$2 π \cdot \frac{2}{2} - \frac{π}{2}$

Combine fractions.

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Combine $2 π$ and $\frac{2}{2}$ .

$\frac{2 π \cdot 2}{2} - \frac{π}{2}$

Combine the numerators over the common denominator.

$\frac{2 π \cdot 2 - π}{2}$

Simplify the numerator.

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Multiply $2$ by $2$ .

$\frac{4 π - π}{2}$

Subtract $π$ from $4 π$ .

$\frac{3 π}{2}$

List the new angles.

$x = \frac{3 π}{2}$

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 20

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 21

Consolidate the answers.

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

Step 22

Verify each of the solutions by substituting them into $\sin (x) + \sqrt{3} \cos (x) = - 1$ and solving.

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