The following question appeared in an examination:
Given that tan x = √3, find the possible values of sin 2x.
A student gave the following answer:
tan x = √3 so x = 60∘ and 2x = 120∘,
therefore sin 2x = √32.
Which one of the following statements is correct?
- A√32 is the only possible value, and this is fully supported by the reasoning given in the student's answer.
- B√32 is the only possible value, but the reasoning given should consider other possible values of x for which tan x = √3.
- C√32 is the only possible value, but the reasoning given should consider other possible values of x for which sin 2x = √32.
- D√32 is not the only possible value because the reasoning given should have considered other possible values of x for which tan x = √3.
- E√32 is not the only possible value because the reasoning given should have considered other possible values of x for which sin 2x = √32.
Show the answer and worked solution
answer · B
- A√32 is the only possible value, and this is fully supported by the reasoning given in the student's answer.
- B√32 is the only possible value, but the reasoning given should consider other possible values of x for which tan x = √3.
- C√32 is the only possible value, but the reasoning given should consider other possible values of x for which sin 2x = √32.
- D√32 is not the only possible value because the reasoning given should have considered other possible values of x for which tan x = √3.
- E√32 is not the only possible value because the reasoning given should have considered other possible values of x for which sin 2x = √32.
The student picked one solution of tan x = √3 without saying why the others do not matter, so the reasoning is incomplete. Checking them: tan has period 180∘, so x = 60∘ + 180∘ n, and doubling gives 2x = 120∘ + 360∘ n, which is the same angle every time. So sin 2x = √32 really is the only value — the answer is right but the argument needs the extra step of ruling the other values of x out.