Basic Math Examples

G = {(\frac{sin (120)}{cos (225)})}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\sin (120)}{\cos (225)})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1

Simplify each term.

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Step 1.1

Simplify the numerator.

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Step 1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

G = {(\frac{sin (60)}{cos (225)})}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\sin (60)}{\cos (225)})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.1.2

The exact value of

sin (60)

$\sin (60)$ is

\frac{\sqrt{3}}{2}

$\frac{\sqrt{3}}{2}$ .

G = {⎛ ⎜ ⎝ \frac{\frac{\sqrt{3}}{2}}{cos (225)} ⎞ ⎟ ⎠}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\frac{\sqrt{3}}{2}}{\cos (225)})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

G = {⎛ ⎜ ⎝ \frac{\frac{\sqrt{3}}{2}}{cos (225)} ⎞ ⎟ ⎠}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\frac{\sqrt{3}}{2}}{\cos (225)})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.2

Simplify the denominator.

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Step 1.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

G = {⎛ ⎜ ⎝ \frac{\frac{\sqrt{3}}{2}}{- cos (45)} ⎞ ⎟ ⎠}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\frac{\sqrt{3}}{2}}{- \cos (45)})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.2.2

The exact value of

cos (45)

$\cos (45)$ is

\frac{\sqrt{2}}{2}

$\frac{\sqrt{2}}{2}$ .

G = {⎛ ⎜ ⎝ \frac{\frac{\sqrt{3}}{2}}{- \frac{\sqrt{2}}{2}} ⎞ ⎟ ⎠}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\frac{\sqrt{3}}{2}}{- \frac{\sqrt{2}}{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

G = {⎛ ⎜ ⎝ \frac{\frac{\sqrt{3}}{2}}{- \frac{\sqrt{2}}{2}} ⎞ ⎟ ⎠}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\frac{\sqrt{3}}{2}}{- \frac{\sqrt{2}}{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

G = {(\frac{\sqrt{3}}{2} (- \frac{2}{\sqrt{2}}))}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\sqrt{3}}{2} (- \frac{2}{\sqrt{2}}))}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.4

Cancel the common factor of

2

$2$ .

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Step 1.4.1

Move the leading negative in

- \frac{2}{\sqrt{2}}

$- \frac{2}{\sqrt{2}}$ into the numerator.

G = {(\frac{\sqrt{3}}{2} \cdot \frac{- 2}{\sqrt{2}})}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\sqrt{3}}{2} \cdot \frac{- 2}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.4.2

Factor

2

$2$ out of

- 2

$- 2$ .

G = {(\frac{\sqrt{3}}{2} \cdot \frac{2 (- 1)}{\sqrt{2}})}^{sec (300)} + \frac{\frac{tan (150)}{sec (210)}}{\frac{csc (120)}{cot (240)}}

$G = {(\frac{\sqrt{3}}{2} \cdot \frac{2 (- 1)}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.4.3

Cancel the common factor.

$G = {(\frac{\sqrt{3}}{2} \cdot \frac{2 \cdot - 1}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.4.4

Rewrite the expression.

$G = {(\sqrt{3} \frac{- 1}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.5

Combine

$\sqrt{3}$ and

$\frac{- 1}{\sqrt{2}}$ .

$G = {(\frac{\sqrt{3} \cdot - 1}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Move

$- 1$ to the left of

$\sqrt{3}$ .

$G = {(\frac{- 1 \cdot \sqrt{3}}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.6.2

Rewrite

$- 1 \sqrt{3}$ as

$- \sqrt{3}$ .

$G = {(\frac{- \sqrt{3}}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.7

Move the negative in front of the fraction.

$G = {(- \frac{\sqrt{3}}{\sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.8

Multiply

$\frac{\sqrt{3}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$G = {(- (\frac{\sqrt{3}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}))}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9

Combine and simplify the denominator.

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Step 1.9.1

Multiply

$\frac{\sqrt{3}}{\sqrt{2}}$ by

$\frac{\sqrt{2}}{\sqrt{2}}$ .

$G = {(- \frac{\sqrt{3} \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.2

Raise

$\sqrt{2}$ to the power of

$1$ .

$G = {(- \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.3

Raise

$\sqrt{2}$ to the power of

$1$ .

$G = {(- \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$G = {(- \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.5

Add

$1$ and

$1$ .

$G = {(- \frac{\sqrt{3} \sqrt{2}}{{\sqrt{2}}^{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.6

Rewrite

${\sqrt{2}}^{2}$ as

$2$ .

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Step 1.9.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{2}$ as

$2^{\frac{1}{2}}$ .

$G = {(- \frac{\sqrt{3} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$G = {(- \frac{\sqrt{3} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$G = {(- \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.6.4

Cancel the common factor of

$2$ .

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Step 1.9.6.4.1

Cancel the common factor.

$G = {(- \frac{\sqrt{3} \sqrt{2}}{2^{\frac{2}{2}}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.6.4.2

Rewrite the expression.

$G = {(- \frac{\sqrt{3} \sqrt{2}}{2^{1}})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.9.6.5

Evaluate the exponent.

$G = {(- \frac{\sqrt{3} \sqrt{2}}{2})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.10

Simplify the numerator.

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Step 1.10.1

Combine using the product rule for radicals.

$G = {(- \frac{\sqrt{3 \cdot 2}}{2})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.10.2

Multiply

$3$ by

$2$ .

$G = {(- \frac{\sqrt{6}}{2})}^{\sec (300)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.11

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$G = {(- \frac{\sqrt{6}}{2})}^{\sec (60)} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.12

The exact value of

$\sec (60)$ is

$2$ .

$G = {(- \frac{\sqrt{6}}{2})}^{2} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.13

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.13.1

Apply the product rule to

$- \frac{\sqrt{6}}{2}$ .

$G = {(- 1)}^{2} {(\frac{\sqrt{6}}{2})}^{2} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.13.2

Apply the product rule to

$\frac{\sqrt{6}}{2}$ .

$G = {(- 1)}^{2} \frac{{\sqrt{6}}^{2}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.14

Raise

$- 1$ to the power of

$2$ .

$G = 1 \frac{{\sqrt{6}}^{2}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.15

Multiply

$\frac{{\sqrt{6}}^{2}}{2^{2}}$ by

$1$ .

$G = \frac{{\sqrt{6}}^{2}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.16

Rewrite

${\sqrt{6}}^{2}$ as

$6$ .

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Step 1.16.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{6}$ as

$6^{\frac{1}{2}}$ .

$G = \frac{{(6^{\frac{1}{2}})}^{2}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.16.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$G = \frac{6^{\frac{1}{2} \cdot 2}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.16.3

Combine

$\frac{1}{2}$ and

$2$ .

$G = \frac{6^{\frac{2}{2}}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.16.4

Cancel the common factor of

$2$ .

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Step 1.16.4.1

Cancel the common factor.

$G = \frac{6^{\frac{2}{2}}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.16.4.2

Rewrite the expression.

$G = \frac{6^{1}}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.16.5

Evaluate the exponent.

$G = \frac{6}{2^{2}} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.17

Raise

$2$ to the power of

$2$ .

$G = \frac{6}{4} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.18

Cancel the common factor of

$6$ and

$4$ .

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Step 1.18.1

Factor

$2$ out of

$6$ .

$G = \frac{2 (3)}{4} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.18.2

Cancel the common factors.

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Step 1.18.2.1

Factor

$2$ out of

$4$ .

$G = \frac{2 \cdot 3}{2 \cdot 2} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.18.2.2

Cancel the common factor.

$G = \frac{2 \cdot 3}{2 \cdot 2} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.18.2.3

Rewrite the expression.

$G = \frac{3}{2} + \frac{\frac{\tan (150)}{\sec (210)}}{\frac{\csc (120)}{\cot (240)}}$

Step 1.19

Multiply the numerator by the reciprocal of the denominator.

$G = \frac{3}{2} + \frac{\tan (150)}{\sec (210)} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.20

Simplify the numerator.

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Step 1.20.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the second quadrant.

$G = \frac{3}{2} + \frac{- \tan (30)}{\sec (210)} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.20.2

The exact value of

$\tan (30)$ is

$\frac{\sqrt{3}}{3}$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{\sec (210)} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21

Simplify the denominator.

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Step 1.21.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because secant is negative in the third quadrant.

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \sec (30)} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.2

The exact value of

$\sec (30)$ is

$\frac{2}{\sqrt{3}}$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2}{\sqrt{3}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.3

Multiply

$\frac{2}{\sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- (\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}})} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4

Combine and simplify the denominator.

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Step 1.21.4.1

Multiply

$\frac{2}{\sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{\sqrt{3} \sqrt{3}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.2

Raise

$\sqrt{3}$ to the power of

$1$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.3

Raise

$\sqrt{3}$ to the power of

$1$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.5

Add

$1$ and

$1$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{{\sqrt{3}}^{2}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.6

Rewrite

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

$3$ .

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Step 1.21.4.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.6.4

Cancel the common factor of

$2$ .

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Step 1.21.4.6.4.1

Cancel the common factor.

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{3^{\frac{2}{2}}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.6.4.2

Rewrite the expression.

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{3^{1}}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.21.4.6.5

Evaluate the exponent.

$G = \frac{3}{2} + \frac{- \frac{\sqrt{3}}{3}}{- \frac{2 \sqrt{3}}{3}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.22

Dividing two negative values results in a positive value.

$G = \frac{3}{2} + \frac{\frac{\sqrt{3}}{3}}{\frac{2 \sqrt{3}}{3}} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.23

Multiply the numerator by the reciprocal of the denominator.

$G = \frac{3}{2} + \frac{\sqrt{3}}{3} \cdot \frac{3}{2 \sqrt{3}} \frac{\cot (240)}{\csc (120)}$

Step 1.24

Cancel the common factor of

$\sqrt{3}$ .

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Step 1.24.1

Factor

$\sqrt{3}$ out of

$2 \sqrt{3}$ .

$G = \frac{3}{2} + \frac{\sqrt{3}}{3} \cdot \frac{3}{\sqrt{3} \cdot 2} \frac{\cot (240)}{\csc (120)}$

Step 1.24.2

Cancel the common factor.

$G = \frac{3}{2} + \frac{\sqrt{3}}{3} \cdot \frac{3}{\sqrt{3} \cdot 2} \frac{\cot (240)}{\csc (120)}$

Step 1.24.3

Rewrite the expression.

$G = \frac{3}{2} + \frac{1}{3} \cdot \frac{3}{2} \frac{\cot (240)}{\csc (120)}$

Step 1.25

Cancel the common factor of

$3$ .

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Step 1.25.1

Cancel the common factor.

$G = \frac{3}{2} + \frac{1}{3} \cdot \frac{3}{2} \frac{\cot (240)}{\csc (120)}$

Step 1.25.2

Rewrite the expression.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{\cot (240)}{\csc (120)}$

Step 1.26

Rewrite

$\csc (120)$ in terms of sines and cosines.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{\cot (240)}{\frac{1}{\sin (120)}}$

Step 1.27

Rewrite

$\cot (240)$ in terms of sines and cosines.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{\frac{\cos (240)}{\sin (240)}}{\frac{1}{\sin (120)}}$

Step 1.28

Multiply by the reciprocal of the fraction to divide by

$\frac{1}{\sin (120)}$ .

$G = \frac{3}{2} + \frac{1}{2} (\frac{\cos (240)}{\sin (240)} \sin (120))$

Step 1.29

Write

$\sin (120)$ as a fraction with denominator

$1$ .

$G = \frac{3}{2} + \frac{1}{2} (\frac{\cos (240)}{\sin (240)} \cdot \frac{\sin (120)}{1})$

Step 1.30

Simplify.

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Step 1.30.1

Divide

$\sin (120)$ by

$1$ .

$G = \frac{3}{2} + \frac{1}{2} (\frac{\cos (240)}{\sin (240)} \sin (120))$

Step 1.30.2

Combine

$\frac{\cos (240)}{\sin (240)}$ and

$\sin (120)$ .

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{\cos (240) \sin (120)}{\sin (240)}$

Step 1.31

Simplify the numerator.

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Step 1.31.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \cos (60) \sin (120)}{\sin (240)}$

Step 1.31.2

The exact value of

$\cos (60)$ is

$\frac{1}{2}$ .

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \frac{1}{2} \sin (120)}{\sin (240)}$

Step 1.31.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \frac{1}{2} \sin (60)}{\sin (240)}$

Step 1.31.4

The exact value of

$\sin (60)$ is

$\frac{\sqrt{3}}{2}$ .

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \frac{1}{2} \cdot \frac{\sqrt{3}}{2}}{\sin (240)}$

Step 1.31.5

Combine exponents.

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Step 1.31.5.1

Multiply

$\frac{\sqrt{3}}{2}$ by

$\frac{1}{2}$ .

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \frac{\sqrt{3}}{2 \cdot 2}}{\sin (240)}$

Step 1.31.5.2

Multiply

$2$ by

$2$ .

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \frac{\sqrt{3}}{4}}{\sin (240)}$

Step 1.32

Simplify the denominator.

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Step 1.32.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \frac{\sqrt{3}}{4}}{- \sin (60)}$

Step 1.32.2

The exact value of

$\sin (60)$ is

$\frac{\sqrt{3}}{2}$ .

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{- \frac{\sqrt{3}}{4}}{- \frac{\sqrt{3}}{2}}$

Step 1.33

Dividing two negative values results in a positive value.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{\frac{\sqrt{3}}{4}}{\frac{\sqrt{3}}{2}}$

Step 1.34

Multiply the numerator by the reciprocal of the denominator.

$G = \frac{3}{2} + \frac{1}{2} (\frac{\sqrt{3}}{4} \cdot \frac{2}{\sqrt{3}})$

Step 1.35

Cancel the common factor of

$\sqrt{3}$ .

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Step 1.35.1

Cancel the common factor.

$G = \frac{3}{2} + \frac{1}{2} (\frac{\sqrt{3}}{4} \cdot \frac{2}{\sqrt{3}})$

Step 1.35.2

Rewrite the expression.

$G = \frac{3}{2} + \frac{1}{2} (\frac{1}{4} \cdot 2)$

Step 1.36

Cancel the common factor of

$2$ .

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Step 1.36.1

Factor

$2$ out of

$4$ .

$G = \frac{3}{2} + \frac{1}{2} (\frac{1}{2 (2)} \cdot 2)$

Step 1.36.2

Cancel the common factor.

$G = \frac{3}{2} + \frac{1}{2} (\frac{1}{2 \cdot 2} \cdot 2)$

Step 1.36.3

Rewrite the expression.

$G = \frac{3}{2} + \frac{1}{2} \cdot \frac{1}{2}$

Step 1.37

Multiply

$\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 1.37.1

Multiply

$\frac{1}{2}$ by

$\frac{1}{2}$ .

$G = \frac{3}{2} + \frac{1}{2 \cdot 2}$

Step 1.37.2

Multiply

$2$ by

$2$ .

$G = \frac{3}{2} + \frac{1}{4}$

Step 2

To write

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

$\frac{2}{2}$ .

$G = \frac{3}{2} \cdot \frac{2}{2} + \frac{1}{4}$

Step 3

Write each expression with a common denominator of

$4$ , by multiplying each by an appropriate factor of

$1$ .

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Step 3.1

Multiply

$\frac{3}{2}$ by

$\frac{2}{2}$ .

$G = \frac{3 \cdot 2}{2 \cdot 2} + \frac{1}{4}$

Step 3.2

Multiply

$2$ by

$2$ .

$G = \frac{3 \cdot 2}{4} + \frac{1}{4}$

Step 4

Combine the numerators over the common denominator.

$G = \frac{3 \cdot 2 + 1}{4}$

Step 5

Simplify the numerator.

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Step 5.1

Multiply

$3$ by

$2$ .

$G = \frac{6 + 1}{4}$

Step 5.2

Add

$6$ and

$1$ .

$G = \frac{7}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$G = \frac{7}{4}$

Decimal Form:

$G = 1.75$

Mixed Number Form:

$G = 1 \frac{3}{4}$