Find the Quadrant of the Angle (tan(175)-tan(55))/(1+tan(175)tan(55))

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Trigonometry

$\frac{\tan (175) - \tan (55)}{1 + \tan (175) \tan (55)}$

Convert the radian measure to degrees.

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To convert radians to degrees, multiply by $\frac{180}{π}$ , since a full circle is $360 °$ or $2 π$ radians.

$(\frac{\tan (175) - \tan (55)}{1 + \tan (175) \tan (55)}) \cdot \frac{180 °}{π}$

Simplify the numerator.

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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.

$\frac{- \tan (5) - \tan (55)}{1 + \tan (175) \tan (55)} \cdot \frac{180}{π}$

Evaluate $\tan (5)$ to get $0.08748866$ .

$\frac{- 1 \cdot 0.08748866 - \tan (55)}{1 + \tan (175) \tan (55)} \cdot \frac{180}{π}$

Multiply $- 1$ by $0.08748866$ to get $- 0.08748866$ .

$\frac{- 0.08748866 - \tan (55)}{1 + \tan (175) \tan (55)} \cdot \frac{180}{π}$

Evaluate $\tan (55)$ to get $1.42814800$ .

$\frac{- 0.08748866 - 1 \cdot 1.42814800}{1 + \tan (175) \tan (55)} \cdot \frac{180}{π}$

Multiply $- 1$ by $1.42814800$ to get $- 1.42814800$ .

$\frac{- 0.08748866 - 1.42814800}{1 + \tan (175) \tan (55)} \cdot \frac{180}{π}$

Subtract $1.42814800$ from $- 0.08748866$ to get $- 1.51563667$ .

$\frac{- 1.51563667}{1 + \tan (175) \tan (55)} \cdot \frac{180}{π}$

Simplify the denominator.

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Remove parentheses.

$\frac{- 1.51563667}{1 + \tan (175) \tan (55)} \cdot \frac{180}{π}$

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.

$\frac{- 1.51563667}{1 - \tan (5) \tan (55)} \cdot \frac{180}{π}$

Evaluate $\tan (5)$ to get $0.08748866$ .

$\frac{- 1.51563667}{1 - 1 \cdot 0.08748866 \tan (55)} \cdot \frac{180}{π}$

Multiply $- 1$ by $0.08748866$ to get $- 0.08748866$ .

$\frac{- 1.51563667}{1 - 0.08748866 \tan (55)} \cdot \frac{180}{π}$

Evaluate $\tan (55)$ to get $1.42814800$ .

$\frac{- 1.51563667}{1 - 0.08748866 \cdot 1.42814800} \cdot \frac{180}{π}$

Multiply $- 0.08748866$ by $1.42814800$ to get $- 0.12494676$ .

$\frac{- 1.51563667}{1 - 0.12494676} \cdot \frac{180}{π}$

Subtract $0.12494676$ from $1$ to get $0.87505323$ .

$\frac{- 1.51563667}{0.87505323} \cdot \frac{180}{π}$

Simplify the expression.

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Divide $- 1.51563667$ by $0.87505323$ to get $- 1.73205080$ .

$- 1.73205080 \cdot \frac{180}{π}$

Multiply $- 1.73205080$ by $\frac{180}{π}$ to get $- 1.73205080 \frac{180}{π}$ .

$- 1.73205080 \frac{180}{π}$

Simplify $- 1.73205080 \frac{180}{π}$ .

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Write $- 1.73205080$ as a fraction with denominator $1$ .

$\frac{- 1.73205080}{1} \frac{180}{π}$

Multiply $\frac{- 1.73205080}{1}$ and $\frac{180}{π}$ to get $\frac{- 1.73205080 \cdot 180}{π}$ .

$\frac{- 1.73205080 \cdot 180}{π}$

Multiply $- 1.73205080$ by $180$ to get $- 311.76914536$ .

$\frac{- 311.76914536}{π}$

Move the negative in front of the fraction.

$- \frac{311.76914536}{π}$

Factor $311.76914536$ out of $311.76914536$ .

$- \frac{311.76914536 (1)}{π}$

Multiply by $1$ .

$- \frac{311.76914536 (1)}{π \cdot 1}$

Separate fractions.

$- (\frac{311.76914536}{π} \frac{1}{1})$

Replace $π$ with an approximation.

$- (\frac{311.76914536}{3.14159265} \frac{1}{1})$

Divide $311.76914536$ by $3.14159265$ to get $99.23920117$ .

$- (99.23920117 (\frac{1}{1}))$

Divide $1$ by $1$ to get $1$ .

$- (99.23920117 \cdot 1)$

Multiply $99.23920117$ by $1$ to get $99.23920117$ .

$- 1 \cdot 99.23920117$

Multiply $- 1$ by $99.23920117$ to get $- 99.23920117$ .

$- 99.23920117$

Convert to a decimal.

$- 99.23920117 °$

For angles smaller than $0 °$ , add $360 °$ to the angle until the angle is larger than $0 °$ .

$260.76079882$

The angle is in the third quadrant.

Quadrant $3$

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