Trigonometry Examples

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

Step 1

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

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

Step 2

Simplify each term.

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Step 2.1

Apply the distributive property.

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

Step 2.2

Multiply $\sqrt{3}$ by $1$ .

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

Step 3

Reorder the polynomial.

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

Step 4

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

$\sqrt{3} (- {(u)}^{2}) - 2 u + \sqrt{3} = 0$

Step 5

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 6

Substitute the values $a = - \sqrt{3}$ , $b = - 2$ , and $c = \sqrt{3}$ into the quadratic formula and solve for $u$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (- \sqrt{3} \cdot \sqrt{3})}}{2 (- \sqrt{3})}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Raise $- 2$ to the power of $2$ .

$u = \frac{2 \pm \sqrt{4 - 4 \cdot (- 1 \sqrt{3}) \cdot \sqrt{3}}}{2 (- \sqrt{3})}$

Step 7.1.2

Multiply $- 4$ by $- 1$ .

$u = \frac{2 \pm \sqrt{4 + 4 \sqrt{3} \cdot \sqrt{3}}}{2 (- \sqrt{3})}$

Step 7.1.3

Multiply $4 \sqrt{3} \sqrt{3}$ .

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Step 7.1.3.1

Raise $\sqrt{3}$ to the power of $1$ .

$u = \frac{2 \pm \sqrt{4 + 4 (\sqrt{3} \sqrt{3})}}{2 (- \sqrt{3})}$

Step 7.1.3.2

Raise $\sqrt{3}$ to the power of $1$ .

$u = \frac{2 \pm \sqrt{4 + 4 (\sqrt{3} \sqrt{3})}}{2 (- \sqrt{3})}$

Step 7.1.3.3

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

$u = \frac{2 \pm \sqrt{4 + 4 {\sqrt{3}}^{1 + 1}}}{2 (- \sqrt{3})}$

Step 7.1.3.4

Add $1$ and $1$ .

$u = \frac{2 \pm \sqrt{4 + 4 {\sqrt{3}}^{2}}}{2 (- \sqrt{3})}$

Step 7.1.4

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

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Step 7.1.4.1

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

$u = \frac{2 \pm \sqrt{4 + 4 {(3^{\frac{1}{2}})}^{2}}}{2 (- \sqrt{3})}$

Step 7.1.4.2

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

$u = \frac{2 \pm \sqrt{4 + 4 \cdot 3^{\frac{1}{2} \cdot 2}}}{2 (- \sqrt{3})}$

Step 7.1.4.3

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

$u = \frac{2 \pm \sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}}{2 (- \sqrt{3})}$

Step 7.1.4.4

Cancel the common factor of $2$ .

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Step 7.1.4.4.1

Cancel the common factor.

$u = \frac{2 \pm \sqrt{4 + 4 \cdot 3^{\frac{2}{2}}}}{2 (- \sqrt{3})}$

Step 7.1.4.4.2

Rewrite the expression.

$u = \frac{2 \pm \sqrt{4 + 4 \cdot 3}}{2 (- \sqrt{3})}$

Step 7.1.4.5

Evaluate the exponent.

$u = \frac{2 \pm \sqrt{4 + 4 \cdot 3}}{2 (- \sqrt{3})}$

Step 7.1.5

Multiply $4$ by $3$ .

$u = \frac{2 \pm \sqrt{4 + 12}}{2 (- \sqrt{3})}$

Step 7.1.6

Add $4$ and $12$ .

$u = \frac{2 \pm \sqrt{16}}{2 (- \sqrt{3})}$

Step 7.1.7

Rewrite $16$ as $4^{2}$ .

$u = \frac{2 \pm \sqrt{4^{2}}}{2 (- \sqrt{3})}$

Step 7.1.8

Pull terms out from under the radical, assuming positive real numbers.

$u = \frac{2 \pm 4}{2 (- \sqrt{3})}$

Step 7.2

Multiply $- 1$ by $2$ .

$u = \frac{2 \pm 4}{- 2 \sqrt{3}}$

Step 7.3

Simplify $\frac{2 \pm 4}{- 2 \sqrt{3}}$ .

$u = \frac{1 \pm 2}{- \sqrt{3}}$

Step 7.4

Move the negative in front of the fraction.

$u = - \frac{1 \pm 2}{\sqrt{3}}$

Step 7.5

Multiply $\frac{1 \pm 2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$u = - (\frac{1 \pm 2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})$

Step 7.6

Combine and simplify the denominator.

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Step 7.6.1

Multiply $\frac{1 \pm 2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$u = - \frac{(1 \pm 2) \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.6.2

Raise $\sqrt{3}$ to the power of $1$ .

$u = - \frac{(1 \pm 2) \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.6.3

Raise $\sqrt{3}$ to the power of $1$ .

$u = - \frac{(1 \pm 2) \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.6.4

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

$u = - \frac{(1 \pm 2) \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.6.5

Add $1$ and $1$ .

$u = - \frac{(1 \pm 2) \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.6.6

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

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Step 7.6.6.1

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

$u = - \frac{(1 \pm 2) \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.6.6.2

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

$u = - \frac{(1 \pm 2) \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.6.6.3

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

$u = - \frac{(1 \pm 2) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.6.4

Cancel the common factor of $2$ .

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Step 7.6.6.4.1

Cancel the common factor.

$u = - \frac{(1 \pm 2) \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.6.4.2

Rewrite the expression.

$u = - \frac{(1 \pm 2) \sqrt{3}}{3}$

Step 7.6.6.5

Evaluate the exponent.

$u = - \frac{(1 \pm 2) \sqrt{3}}{3}$

Step 7.7

Reorder factors in $- \frac{(1 \pm 2) \sqrt{3}}{3}$ .

$u = - \frac{\sqrt{3} (1 \pm 2)}{3}$

Step 8

The final answer is the combination of both solutions.

$u = - \sqrt{3}, \frac{\sqrt{3}}{3}$

Step 9

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

$\sin (x) = - \sqrt{3}, \frac{\sqrt{3}}{3}$

Step 10

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

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

$\sin (x) = \frac{\sqrt{3}}{3}$

Step 11

Solve for $x$ in $\sin (x) = - \sqrt{3}$ .

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Step 11.1

The range of sine is $- 1 \leq y \leq 1$ . Since $- \sqrt{3}$ does not fall in this range, there is no solution.

No solution

Step 12

Solve for $x$ in $\sin (x) = \frac{\sqrt{3}}{3}$ .

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Step 12.1

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

$x = \arcsin (\frac{\sqrt{3}}{3})$

Step 12.2

Simplify the right side.

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Step 12.2.1

Evaluate $\arcsin (\frac{\sqrt{3}}{3})$ .

$x = 0.6154797$

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.

$x = (3.14159265) - 0.6154797$

Step 12.4

Solve for $x$ .

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Step 12.4.1

Remove parentheses.

$x = 3.14159265 - 0.6154797$

Step 12.4.2

Remove parentheses.

$x = (3.14159265) - 0.6154797$

Step 12.4.3

Subtract $0.6154797$ from $3.14159265$ .

$x = 2.52611294$

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.

$x = 0.6154797 + 2 π n, 2.52611294 + 2 π n$ , for any integer $n$

Step 13

List all of the solutions.

$x = 0.6154797 + 2 π n, 2.52611294 + 2 π n$ , for any integer $n$