et z1𝑧1 and z2𝑧2 be the nth𝑛𝑡ℎ roots of unity which are ends of a line segment that subtend a right angle at the origin. Then, n𝑛 must be of the form
Question
Let and be the th roots of unity which are ends of a line segment that subtend a right angle at the origin. Then, must be of the form
Solution
The nth roots of unity are given by cos(2πk/n) + isin(2πk/n) for k = 0, 1, ..., n-1.
If z1 and z2 subtend a right angle at the origin, then the angle between them is 90 degrees or π/2 radians.
This means that the difference in the arguments of z1 and z2 is π/2.
In terms of k, this means that (2π/n)(k2 - k1) = π/2, where k1 and k2 are the values of k corresponding to z1 and z2.
Solving for n, we get n = 4(k2 - k1).
Since k2 and k1 are integers, n must be a multiple of 4.
Therefore, n must be of the form 4m, where m is an integer.
Similar Questions
et z1𝑧1 and z2𝑧2 be the nth𝑛𝑡ℎ roots of unity which are ends of a line segment that subtend a right angle at the origin. Then, n𝑛 must be of the form
is a chord of a circle with centre O : A is a point on an minor arc BC. Then ∠BAC − ∠OBC is equal to
The figure shown has two parallel lines cut by a transversal:Which angle is a corresponding angle to ∠6?Group of answer choices∠7∠8∠3∠2
Two lines intersecting at a right angle form a line. are parallel. are perpendicular. form a ray.
Line C D intersects line A B at point P. What is true about angle ∠APD? It is to angle Angle A P C. It is ✔ vertical to angle Angle C P B.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.