Discussion Overview
The discussion revolves around expressing the trigonometric functions cos(2 tan^-1(x/4)) and sin(2 tan^-1(x/4)) as algebraic expressions in terms of x. Participants are exploring the relationships between the trigonometric identities and the inverse tangent function, with a focus on deriving these expressions correctly.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant presents their derived expressions for cos(2 tan^-1(x/4)) and sin(2 tan^-1(x/4)) as cos(theta) = 8*sqrt(x^2+64)/(x^2+64) and sin(theta) = x*sqrt(x^2+64)/(x^2+64).
- Another participant suggests using the identities cos(2t) = 2cos^2(t) - 1 and sin(2t) = 2sin(t)cos(t) after defining t = tan^-1(x/4), providing expressions for sin(t) and cos(t) in terms of x.
- A participant questions the interpretation of 2*tan^-1(x/4), suggesting a potential simplification to x/8, which is challenged by another participant who clarifies the distinction between the angle and its double.
Areas of Agreement / Disagreement
Participants express differing interpretations of the relationship between the angles and their tangent functions, leading to a lack of consensus on the correct approach to derive the expressions.
Contextual Notes
Some participants rely on specific trigonometric identities and relationships, which may depend on the correct interpretation of the angles involved. There are unresolved steps in the derivation process, particularly regarding the manipulation of the inverse tangent function.