If y = xtan(11π/24) and y = xtan(19π/24) represents two straight lines at right angles,
prove that the angle between the axis is π/4.
Gud
To prove that the angle between the axes is π/4, we need to show that the lines y = xtan(11π/24) and y = xtan(19π/24) are perpendicular.
Let's find the slopes of these lines. The slope of a line with angle θ to the x-axis can be represented as tan(θ). In this case, the slopes of the two lines are tan(11π/24) and tan(19π/24), respectively.
To check if two lines are perpendicular, we need to verify if the product of their slopes is -1. So, let's calculate the product of the slopes:
tan(11π/24) * tan(19π/24)
To calculate this product, we need to use the trigonometric identity: tan(θ) * tan(π/2 - θ) = 1
Using this identity, we can rewrite the product as:
tan(11π/24) * tan(π/2 - 11π/24)
Simplifying further:
tan(11π/24) * tan(13π/24)
Now, let's calculate the values of these tangents. We can use a calculator or trigonometric identities to find them.
tan(11π/24) ≈ 0.57735
tan(13π/24) ≈ 1.73205
So, the product of the slopes is:
0.57735 * 1.73205 ≈ 1
Since the product of the slopes is 1 and not -1, the two lines are not perpendicular. Therefore, the angle between the axes is not π/4.