Trigonometry Examples

$v = \sqrt{\frac{9.8 \cdot 83^{2}}{83 \tan (30 °) - 40} \cdot \frac{1}{2 \cos^{2} (30 °)}}$

Step 1

Multiply $\frac{9.8 \cdot 83^{2}}{83 \tan (30 °) - 40}$ by $\frac{1}{2 \cos^{2} (30 °)}$ .

$v = \sqrt{\frac{9.8 \cdot 83^{2}}{(83 \tan (30 °) - 40) (2 \cos^{2} (30 °))}}$

Step 2

The exact value of $\tan (30 °)$ is $\frac{\sqrt{3}}{3}$ .

$v = \sqrt{\frac{9.8 \cdot 83^{2}}{(83 \frac{\sqrt{3}}{3} - 40) (2 \cos^{2} (30 °))}}$

Step 3

Combine $83$ and $\frac{\sqrt{3}}{3}$ .

$v = \sqrt{\frac{9.8 \cdot 83^{2}}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \cos^{2} (30 °)}}$

Step 4

Multiply $9.8$ by $83^{2}$ .

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \cos^{2} (30 °)}}$

Step 5

Simplify the denominator.

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Step 5.1

The exact value of $\cos (30 °)$ is $\frac{\sqrt{3}}{2}$ .

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 {(\frac{\sqrt{3}}{2})}^{2}}}$

Step 5.2

Apply the product rule to $\frac{\sqrt{3}}{2}$ .

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{{\sqrt{3}}^{2}}{2^{2}}}}$

Step 5.3

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

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Step 5.3.1

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

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{{(3^{\frac{1}{2}})}^{2}}{2^{2}}}}$

Step 5.3.2

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

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{3^{\frac{1}{2} \cdot 2}}{2^{2}}}}$

Step 5.3.3

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

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{3^{\frac{2}{2}}}{2^{2}}}}$

Step 5.3.4

Cancel the common factor of $2$ .

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Step 5.3.4.1

Cancel the common factor.

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{3^{\frac{2}{2}}}{2^{2}}}}$

Step 5.3.4.2

Rewrite the expression.

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{3^{1}}{2^{2}}}}$

Step 5.3.5

Evaluate the exponent.

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{3}{2^{2}}}}$

Step 5.4

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

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot 2 \frac{3}{4}}}$

Step 5.5

Combine exponents.

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Step 5.5.1

Combine $\frac{3}{4}$ and $2$ .

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot \frac{3 \cdot 2}{4}}}$

Step 5.5.2

Multiply $3$ by $2$ .

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40) \cdot \frac{6}{4}}}$

Step 5.6

To write $- 40$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} - 40 \cdot \frac{3}{3}) \cdot \frac{6}{4}}}$

Step 5.7

Combine $- 40$ and $\frac{3}{3}$ .

$v = \sqrt{\frac{67512.2}{(\frac{83 \sqrt{3}}{3} + \frac{- 40 \cdot 3}{3}) \cdot \frac{6}{4}}}$

Step 5.8

Combine the numerators over the common denominator.

$v = \sqrt{\frac{67512.2}{\frac{83 \sqrt{3} - 40 \cdot 3}{3} \cdot \frac{6}{4}}}$

Step 5.9

Multiply $- 40$ by $3$ .

$v = \sqrt{\frac{67512.2}{\frac{83 \sqrt{3} - 120}{3} \cdot \frac{6}{4}}}$

Step 5.10

Cancel the common factor of $6$ and $4$ .

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Step 5.10.1

Factor $2$ out of $6$ .

$v = \sqrt{\frac{67512.2}{\frac{83 \sqrt{3} - 120}{3} \cdot \frac{2 (3)}{4}}}$

Step 5.10.2

Cancel the common factors.

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Step 5.10.2.1

Factor $2$ out of $4$ .

$v = \sqrt{\frac{67512.2}{\frac{83 \sqrt{3} - 120}{3} \cdot \frac{2 \cdot 3}{2 \cdot 2}}}$

Step 5.10.2.2

Cancel the common factor.

$v = \sqrt{\frac{67512.2}{\frac{83 \sqrt{3} - 120}{3} \cdot \frac{2 \cdot 3}{2 \cdot 2}}}$

Step 5.10.2.3

Rewrite the expression.

$v = \sqrt{\frac{67512.2}{\frac{83 \sqrt{3} - 120}{3} \cdot \frac{3}{2}}}$

Step 6

Multiply $\frac{83 \sqrt{3} - 120}{3}$ by $\frac{3}{2}$ .

$v = \sqrt{\frac{67512.2}{\frac{(83 \sqrt{3} - 120) \cdot 3}{3 \cdot 2}}}$

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Multiply $3$ by $2$ .

$v = \sqrt{\frac{67512.2}{\frac{(83 \sqrt{3} - 120) \cdot 3}{6}}}$

Step 7.2

Reduce the expression $\frac{(83 \sqrt{3} - 120) \cdot 3}{6}$ by cancelling the common factors.

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Step 7.2.1

Factor $3$ out of $(83 \sqrt{3} - 120) \cdot 3$ .

$v = \sqrt{\frac{67512.2}{\frac{3 \cdot (83 \sqrt{3} - 120)}{6}}}$

Step 7.2.2

Factor $3$ out of $6$ .

$v = \sqrt{\frac{67512.2}{\frac{3 \cdot (83 \sqrt{3} - 120)}{3 \cdot 2}}}$

Step 7.2.3

Cancel the common factor.

$v = \sqrt{\frac{67512.2}{\frac{3 \cdot (83 \sqrt{3} - 120)}{3 \cdot 2}}}$

Step 7.2.4

Rewrite the expression.

$v = \sqrt{\frac{67512.2}{\frac{83 \sqrt{3} - 120}{2}}}$

Step 8

Multiply the numerator by the reciprocal of the denominator.

$v = \sqrt{67512.2 \frac{2}{83 \sqrt{3} - 120}}$

Step 9

Multiply $\frac{2}{83 \sqrt{3} - 120}$ by $\frac{83 \sqrt{3} + 120}{83 \sqrt{3} + 120}$ .

$v = \sqrt{67512.2 (\frac{2}{83 \sqrt{3} - 120} \cdot \frac{83 \sqrt{3} + 120}{83 \sqrt{3} + 120})}$

Step 10

Multiply $\frac{2}{83 \sqrt{3} - 120}$ by $\frac{83 \sqrt{3} + 120}{83 \sqrt{3} + 120}$ .

$v = \sqrt{67512.2 \frac{2 (83 \sqrt{3} + 120)}{(83 \sqrt{3} - 120) (83 \sqrt{3} + 120)}}$

Step 11

Expand the denominator using the FOIL method.

$v = \sqrt{67512.2 \frac{2 (83 \sqrt{3} + 120)}{6889 {\sqrt{3}}^{2} + 9960 \sqrt{3} - 9960 \sqrt{3} - 14400}}$

Step 12

Simplify.

$v = \sqrt{67512.2 \frac{2 (83 \sqrt{3} + 120)}{6267}}$

Step 13

Multiply $67512.2 \frac{2 (83 \sqrt{3} + 120)}{6267}$ .

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Step 13.1

Combine $67512.2$ and $\frac{2 (83 \sqrt{3} + 120)}{6267}$ .

$v = \sqrt{\frac{67512.2 (2 (83 \sqrt{3} + 120))}{6267}}$

Step 13.2

Multiply $2$ by $67512.2$ .

$v = \sqrt{\frac{135024.4 (83 \sqrt{3} + 120)}{6267}}$

Step 13.3

Multiply $135024.4$ by $83 \sqrt{3} + 120$ .

$v = \sqrt{\frac{35614065.04810478}{6267}}$

Step 14

Divide $35614065.04810478$ by $6267$ .

$v = \sqrt{5682.79321016}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$v = \sqrt{5682.79321016}$

Decimal Form:

$v = 75.38430347 \dots$