Simplify complex fraction ( ( 2x - 5 ) / ( x^2 - 9) ) / ( ( 3x - 1 ) / ( x + 3 ) )

Mathematics

2 answers:

Dovator [93]3 years ago

4 0

Hope this helped! Ask me if there's any part of the working you don't understand! :)

sukhopar [10]3 years ago

4 0

<span>( ( 2x - 5 ) / ( x^2 - 9) ) / ( ( 3x - 1 ) / ( x + 3 ) )=

=</span>( ( 2x - 5 ) / ( x-3)(x+3) ) / ( ( 3x - 1 ) / ( x + 3 ) )=

= ( ( 2x - 5 ) / ( x-3) / ( ( 3x - 1 ) )=( 2x - 5 ) ( 3x - 1 )/( x-3)

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Consider the absolute value, because we only worry about the quadrant later.

$\text{Consider: } sin(A) = |-\frac{\sqrt{3}}{4}|$
$sin(A) = \frac{\sqrt{3}}{4}$

Thus, we know that the hypotenuse has a length of 4 units, and the side opposite the angle, A is √3, because this is the nature of the sine function in relation to its triangular component.

The missing side can be found using Pythagoras' Theorem:
4² - (√3)² = x²
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13 = x²
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$\therefore tan(A) = \pm \sqrt{\frac{3}{13}}$

Since angle A is in the third quadrant, the tangent function will produce a positive angle.

$tan(A) = \sqrt{\frac{3}{13}}$
$\text{Rationalise the denominator: } \frac{\sqrt{3}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}$

$tan(A) = \frac{\sqrt{39}}{13}$