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A line has equation $y = rac{ ext{√}3}{x} + 5 = 0.$ Determine the angle this line makes with the positive direction of the x-axis. - Scottish Highers Maths - Question 8 - 2018

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A-line-has-equation-$y-=--rac{-ext{√}3}{x}-+-5-=-0.$---Determine-the-angle-this-line-makes-with-the-positive-direction-of-the-x-axis.-Scottish Highers Maths-Question 8-2018.png

A line has equation $y = rac{ ext{√}3}{x} + 5 = 0.$ Determine the angle this line makes with the positive direction of the x-axis.

Worked Solution & Example Answer:A line has equation $y = rac{ ext{√}3}{x} + 5 = 0.$ Determine the angle this line makes with the positive direction of the x-axis. - Scottish Highers Maths - Question 8 - 2018

Step 1

determine the gradient of the line

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Answer

To determine the angle a line makes with the x-axis, we first need to find the gradient (slope) of the line. Rewriting the equation:

y = - rac{ ext{√}3}{1}x - 5

From this, we can observe that the gradient mm is −ext√3- ext{√}3.

Step 2

determine the angle

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Answer

The angle heta heta the line makes with the positive direction of the x-axis can be calculated using the tangent function:

an(heta)=extGradient=m=−ext√3 an( heta) = ext{Gradient} = m = - ext{√}3

To find the angle:

heta=an−1(−ext√3) heta = an^{-1}(- ext{√}3)

The angle corresponding to −ext√3- ext{√}3 is 180exto−60exto=120exto180^ ext{o} - 60^ ext{o} = 120^ ext{o}. Therefore, the angle the line makes with the positive direction of the x-axis is 120exto120^ ext{o}.

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