Question

What angle does the line y= -X + 2 3 make with the positive side of the line y=22 OU ° (Type your answer in degrees.) 4

Answer 1

We will investigate how to determine an angle formed by a straight line relative to any horizontal line plot on a two dimensional cartesian coordinate grid.

The following two equation of lines are given:

`\begin{gathered} y\text{ = }\frac{\sqrt[]{3}}{3}\cdot x\text{ + }2\text{ }\ldots Eq1 \\ \\ y\text{ = 2 }\ldots\text{ Eq2} \end{gathered}`

All straight lines are expressed in the standard general form of an equation of a line:

`y=\text{ m}\cdot x\text{ + c}`

Where,

`\begin{gathered} m\colon\text{ slope/gradient} \\ c\colon\text{ y-intercept} \end{gathered}`

To determine an angle we must always deal with the reference equation of a line. The reference equation of a line is a horizontal line parallel to the ( x-axis ). So the anlge formed between the (Eq1) and (Eq2) is equivalent to the angle formed between (Eq1) and the x-axis.

To determine the angle of any line with respect to the x-axis. We seek help from trigonometric ratios. There are three type of trigonometric ratios as follows:

`\sin \text{ ( }\theta\text{ ) = }(P)/(H)\text{ , cos ( }\theta\text{ ) = }(B)/(H)\text{ , tan ( }\theta\text{ ) = }(P)/(B)`

Where,

`\begin{gathered} H\colon\text{ Hypotenuse of the right angle formed} \\ P\colon\text{ Side opposite to the angle ( }\theta\text{ )} \\ B\colon\text{ Side adjacent to the }angle\text{ ( }\theta\text{ )} \end{gathered}`

For sine and cosine trigonometric ratio we will need to evaluate the length of the line ( H ). This can be a tedious process. However, for the case of tangent ratio we need the sides ( P and B ).

The sides ( P ) and ( B ) can be denoted as ( rise ) and ( run ) of the straight line. The ratio of rise over run is also expressed by the parameter of the equation of straight line as follows:

`m\text{ = }(rise)/(run)\text{ = }(y_2-y_1)/(x_2-x_1)\text{ = }(P)/(B)`

The above relation always holds true for ALL straight lines!

We can therefore modify our tangent ratio as such:

`\begin{gathered} \tan \text{ ( }\theta\text{ ) = }(P)/(B)\text{ , then:} \\ \\ \tan \text{ ( }\theta\text{ ) = m }\ldots\text{ Eq3} \end{gathered}`

The (Eq3) is almost always used to determine the angle between any line (Eq1) and any horizontal line like ( Eq2 ). Its a standard result!

We will use the ( Eq3 ) to determine the angle formed as follows:

`\tan (\theta)\text{ = }\frac{\sqrt[]{3}}{3},\text{ ( m ) taken from Eq1}`

Next we recall the standard angle table for all trigonometric ratios as shown below:

From the above table we can see the value of angle denoted in the top row and the corresponding ratios for each trigonometric function.

Locate the tan ( theta ) row-4,column-1. Then locate the trigonometric ratio Column-3!

Then read out the angle at Row-1 and column-3. The angle defined is:

`\theta\text{ = 30 degrees}`

Hence, the angle between the given two lines is:

`30\text{ degrees}`