Question Video: Finding the Angle between Two Straight Lines in Two Dimensions Mathematics • First Year of Secondary School

Video Transcript

Determine the measure of the acute angle between the straight line 𝑥 minus 𝑦 plus four equals zero and the straight line passing through the points three, negative two and negative two, four to the nearest second.

So we’re asked to determine the measure of the acute angle between two straight lines. We have a formula that we can use to work this out. Tan of the angle 𝜃 is equal to the modulus of 𝑚 one minus 𝑚 two over one plus 𝑚 one 𝑚 two, where 𝑚 one and 𝑚 two represent the slopes of the two lines. We therefore need to find the slopes of the two lines so we can substitute the values of 𝑚 one and 𝑚 two into this formula.

Let’s start with the line 𝑥 minus 𝑦 plus four is equal to zero. If we add 𝑦 to both sides of this equation, we get the equation 𝑥 plus four is equal to 𝑦. Or equivalently, 𝑦 is equal to 𝑥 plus four. Comparing this with the slope intercept form of the equation of a straight line, 𝑦 equals 𝑚𝑥 plus 𝑐, we can see that the slope of this line is equal to one.

Next, we need to find the slope of line two. We’re told the coordinates of two points that lie on this line. If we know two points that lie on a line, 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, then the slope can be calculated as the change in 𝑦 divided by the change in 𝑥: 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. Substituting the coordinates of the two points on this straight line, we have that 𝑦 two minus 𝑦 one is equal to four minus negative two. And 𝑥 two minus 𝑥 one is equal to negative two minus three. This simplifies to six over negative five, which is more usually written as negative six over five.

So now that we know the slopes of the two lines, we can substitute these values into our formula for calculating the angle between them. We have that tan of 𝜃 is equal to the modulus of one minus negative six over five divided by one plus one multiplied by negative six over five. Now this can be simplified if we think of one as five over five.

In the numerator, we have five over five minus negative six over five, which becomes 11 over five, and in the denominator five over five minus six over five, which is negative one over five. The denominators of five in both the numerator and denominator of the overall fraction will cancel each other out, leaving the modulus of negative eleven, which is just 11.

To find the angle 𝜃, we need to use the inverse tan function. We have that 𝜃 is equal to inverse tan of 11, which is 84.80557 degrees. The question asked for this angle to the nearest second. So we need to convert from degrees to degrees, minutes, and seconds.

Your calculator may have a function that does that for you automatically. But if not, then you need to remember that there are 60 minutes in a degree and 60 seconds in a minute and use that to convert the decimal part of this answer into minutes and seconds. It gives an answer of 84 degrees, 48 minutes, and 20 seconds to the nearest second.