Trigonometry Examples

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

Step 1

Simplify $(2 \sin (x) - \sqrt{3}) (3 \sec (x) + 2 \sqrt{3})$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Combine $3$ and $\frac{1}{\cos (x)}$ .

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

Step 1.2

Expand $(2 \sin (x) - \sqrt{3}) (\frac{3}{\cos (x)} + 2 \sqrt{3})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $2 \sin (x) \frac{3}{\cos (x)}$ .

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Step 1.3.1.1.1

Combine $\frac{3}{\cos (x)}$ and $2$ .

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

Step 1.3.1.1.2

Multiply $3$ by $2$ .

$\frac{6}{\cos (x)} \sin (x) + 2 \sin (x) (2 \sqrt{3}) - \sqrt{3} \frac{3}{\cos (x)} - \sqrt{3} (2 \sqrt{3}) = 0$

Step 1.3.1.1.3

Combine $\frac{6}{\cos (x)}$ and $\sin (x)$ .

$\frac{6 \sin (x)}{\cos (x)} + 2 \sin (x) (2 \sqrt{3}) - \sqrt{3} \frac{3}{\cos (x)} - \sqrt{3} (2 \sqrt{3}) = 0$

Step 1.3.1.2

Multiply $2$ by $2$ .

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \sqrt{3} \frac{3}{\cos (x)} - \sqrt{3} (2 \sqrt{3}) = 0$

Step 1.3.1.3

Combine $\frac{3}{\cos (x)}$ and $\sqrt{3}$ .

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - \sqrt{3} (2 \sqrt{3}) = 0$

Step 1.3.1.4

Multiply $- \sqrt{3} (2 \sqrt{3})$ .

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Step 1.3.1.4.1

Multiply $2$ by $- 1$ .

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 \sqrt{3} \sqrt{3} = 0$

Step 1.3.1.4.2

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

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 ({\sqrt{3}}^{1} \sqrt{3}) = 0$

Step 1.3.1.4.3

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

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) = 0$

Step 1.3.1.4.4

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

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 {\sqrt{3}}^{1 + 1} = 0$

Step 1.3.1.4.5

Add $1$ and $1$ .

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 {\sqrt{3}}^{2} = 0$

Step 1.3.1.5

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

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Step 1.3.1.5.1

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

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 {(3^{\frac{1}{2}})}^{2} = 0$

Step 1.3.1.5.2

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

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 \cdot 3^{\frac{1}{2} \cdot 2} = 0$

Step 1.3.1.5.3

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

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 \cdot 3^{\frac{2}{2}} = 0$

Step 1.3.1.5.4

Cancel the common factor of $2$ .

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Step 1.3.1.5.4.1

Cancel the common factor.

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 \cdot 3^{\frac{2}{2}} = 0$

Step 1.3.1.5.4.2

Rewrite the expression.

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 \cdot 3^{1} = 0$

Step 1.3.1.5.5

Evaluate the exponent.

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 2 \cdot 3 = 0$

Step 1.3.1.6

Multiply $- 2$ by $3$ .

$\frac{6 \sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 6 = 0$

Step 1.3.2

Simplify each term.

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Step 1.3.2.1

Separate fractions.

$\frac{6}{1} \cdot \frac{\sin (x)}{\cos (x)} + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 6 = 0$

Step 1.3.2.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{6}{1} \tan (x) + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 6 = 0$

Step 1.3.2.3

Divide $6$ by $1$ .

$6 \tan (x) + 4 \sin (x) \sqrt{3} - \frac{3 \sqrt{3}}{\cos (x)} - 6 = 0$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = \frac{π}{3} + 2 π n, \frac{2 π}{3} + 2 π n, \frac{5 π}{6} + 2 π n, \frac{7 π}{6} + 2 π n$ , for any integer $n$

Step 3