Trigonometry Examples

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

Step 1

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

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

Step 2

Square both sides of the equation.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

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

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

Step 3.2.3

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

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

Step 3.2.4

Cancel the common factor of $2$ .

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Step 3.2.4.1

Cancel the common factor.

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

Step 3.2.4.2

Rewrite the expression.

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

Step 3.2.5

Evaluate the exponent.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Apply the distributive property.

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

Step 4.2.2

Apply the distributive property.

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

Step 4.2.3

Apply the distributive property.

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

Step 4.3

Simplify and combine like terms.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Multiply $1$ by $1$ .

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

Step 4.3.1.2

Multiply $- \cos (x)$ by $1$ .

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

Step 4.3.1.3

Multiply $- 1$ by $1$ .

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

Step 4.3.1.4

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

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Step 4.3.1.4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.1.4.2

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

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

Step 4.3.1.4.3

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

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

Step 4.3.1.4.4

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

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

Step 4.3.1.4.5

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

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

Step 4.3.1.4.6

Add $1$ and $1$ .

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

Step 4.3.2

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

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

Step 5

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

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Step 5.1

Subtract $1$ from both sides of the equation.

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

Step 5.2

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

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

Step 5.3

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

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

Step 6

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

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

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

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

Step 7.2

Multiply $3$ by $1$ .

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

Step 7.3

Multiply $- 1$ by $3$ .

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

Step 8

Simplify by adding terms.

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Step 8.1

Subtract $1$ from $3$ .

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

Step 8.2

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

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

Step 9

Reorder the polynomial.

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

Step 10

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

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

Step 11

Factor the left side of the equation.

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Step 11.1

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

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Step 11.1.1

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

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

Step 11.1.2

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

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

Step 11.1.3

Factor $- 2$ out of $2$ .

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

Step 11.1.4

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

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

Step 11.1.5

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

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

Step 11.2

Factor.

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Step 11.2.1

Factor by grouping.

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Step 11.2.1.1

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 11.2.1.1.1

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

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

Step 11.2.1.1.2

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

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

Step 11.2.1.1.3

Apply the distributive property.

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

Step 11.2.1.1.4

Multiply $u$ by $1$ .

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

Step 11.2.1.2

Factor out the greatest common factor from each group.

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Step 11.2.1.2.1

Group the first two terms and the last two terms.

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

Step 11.2.1.2.2

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

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

Step 11.2.1.3

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

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

Step 11.2.2

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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Step 13.1

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

$2 u + 1 = 0$

Step 13.2

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

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Step 13.2.1

Subtract $1$ from both sides of the equation.

$2 u = - 1$

Step 13.2.2

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

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Step 13.2.2.1

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

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

Step 13.2.2.2

Simplify the left side.

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Step 13.2.2.2.1

Cancel the common factor of $2$ .

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Step 13.2.2.2.1.1

Cancel the common factor.

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

Step 13.2.2.2.1.2

Divide $u$ by $1$ .

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

Step 13.2.2.3

Simplify the right side.

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Step 13.2.2.3.1

Move the negative in front of the fraction.

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

Step 14

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

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Step 14.1

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

$u - 1 = 0$

Step 14.2

Add $1$ to 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 $\cos (x)$ for $u$ .

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

Step 17

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

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

$\cos (x) = 1$

Step 18

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

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Step 18.1

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

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

Step 18.2

Simplify the right side.

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Step 18.2.1

The exact value of $\arccos (- \frac{1}{2})$ is $\frac{2 π}{3}$ .

$x = \frac{2 π}{3}$

Step 18.3

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.

$x = 2 π - \frac{2 π}{3}$

Step 18.4

Simplify $2 π - \frac{2 π}{3}$ .

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Step 18.4.1

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

$x = 2 π \cdot \frac{3}{3} - \frac{2 π}{3}$

Step 18.4.2

Combine fractions.

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Step 18.4.2.1

Combine $2 π$ and $\frac{3}{3}$ .

$x = \frac{2 π \cdot 3}{3} - \frac{2 π}{3}$

Step 18.4.2.2

Combine the numerators over the common denominator.

$x = \frac{2 π \cdot 3 - 2 π}{3}$

Step 18.4.3

Simplify the numerator.

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Step 18.4.3.1

Multiply $3$ by $2$ .

$x = \frac{6 π - 2 π}{3}$

Step 18.4.3.2

Subtract $2 π$ from $6 π$ .

$x = \frac{4 π}{3}$

Step 18.5

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

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Step 18.5.1

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

$\frac{2 π}{| b |}$

Step 18.5.2

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

$\frac{2 π}{| 1 |}$

Step 18.5.3

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

$\frac{2 π}{1}$

Step 18.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 18.6

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

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n$ , for any integer $n$

Step 19

Solve for $x$ in $\cos (x) = 1$ .

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Step 19.1

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

$x = \arccos (1)$

Step 19.2

Simplify the right side.

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Step 19.2.1

The exact value of $\arccos (1)$ is $0$ .

$x = 0$

Step 19.3

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

$x = 2 π - 0$

Step 19.4

Subtract $0$ from $2 π$ .

$x = 2 π$

Step 19.5

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

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Step 19.5.1

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

$\frac{2 π}{| b |}$

Step 19.5.2

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

$\frac{2 π}{| 1 |}$

Step 19.5.3

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

$\frac{2 π}{1}$

Step 19.5.4

Divide $2 π$ by $1$ .

$2 π$

Step 19.6

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

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

Step 20

List all of the solutions.

$x = \frac{2 π}{3} + 2 π n, \frac{4 π}{3} + 2 π n, 2 π n, 2 π + 2 π n$ , for any integer $n$

Step 21

Consolidate the answers.

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

Step 22

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

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