Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.1.2

Multiply $\frac{\cos (x)}{\sin (x)} \cos (x)$ .

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Step 1.1.2.1

Combine $\frac{\cos (x)}{\sin (x)}$ and $\cos (x)$ .

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

Step 1.1.2.2

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

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

Step 1.1.2.3

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

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

Step 1.1.2.4

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

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

Step 1.1.2.5

Add $1$ and $1$ .

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

Step 2

Multiply both sides of the equation by $\sin (x)$ .

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

Step 3

Apply the distributive property.

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

Add $1$ and $1$ .

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

Step 5

Cancel the common factor of $\sin (x)$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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

Step 6

Apply pythagorean identity.

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

Step 7

Rewrite the equation as $\sin (x) \sqrt{3} = 1$ .

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

Step 8

Divide each term in $\sin (x) \sqrt{3} = 1$ by $\sqrt{3}$ and simplify.

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Step 8.1

Divide each term in $\sin (x) \sqrt{3} = 1$ by $\sqrt{3}$ .

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

Step 8.2

Simplify the left side.

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Step 8.2.1

Cancel the common factor of $\sqrt{3}$ .

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Step 8.2.1.1

Cancel the common factor.

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

Step 8.2.1.2

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

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

Step 8.3

Simplify the right side.

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Step 8.3.1

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

$\sin (x) = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 8.3.2

Combine and simplify the denominator.

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Step 8.3.2.1

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

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

Step 8.3.2.2

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

$\sin (x) = \frac{\sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 8.3.2.3

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

$\sin (x) = \frac{\sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 8.3.2.4

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

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

Step 8.3.2.5

Add $1$ and $1$ .

$\sin (x) = \frac{\sqrt{3}}{{\sqrt{3}}^{2}}$

Step 8.3.2.6

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

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Step 8.3.2.6.1

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

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

Step 8.3.2.6.2

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

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

Step 8.3.2.6.3

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

$\sin (x) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.2.6.4

Cancel the common factor of $2$ .

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Step 8.3.2.6.4.1

Cancel the common factor.

$\sin (x) = \frac{\sqrt{3}}{3^{\frac{2}{2}}}$

Step 8.3.2.6.4.2

Rewrite the expression.

$\sin (x) = \frac{\sqrt{3}}{3^{1}}$

Step 8.3.2.6.5

Evaluate the exponent.

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

Step 9

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

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

Step 10

Simplify the right side.

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Step 10.1

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

$x = 0.6154797$

Step 11

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

Solve for $x$ .

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Step 12.1

Remove parentheses.

$x = 3.14159265 - 0.6154797$

Step 12.2

Remove parentheses.

$x = (3.14159265) - 0.6154797$

Step 12.3

Subtract $0.6154797$ from $3.14159265$ .

$x = 2.52611294$

Step 13

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

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Step 13.1

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

$\frac{2 π}{| b |}$

Step 13.2

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

$\frac{2 π}{| 1 |}$

Step 13.3

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

$\frac{2 π}{1}$

Step 13.4

Divide $2 π$ by $1$ .

$2 π$

Step 14

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 $n$