Discussion Overview
The discussion revolves around calculating the value of $\cos\left({\sin^{-1}\left({2/3}\right)}\right)$, exploring the relationships between trigonometric functions, specifically sine, cosine, and tangent, in the context of inverse trigonometric functions. The scope includes mathematical reasoning and technical explanations related to trigonometric identities and their applications.
Discussion Character
- Mathematical reasoning
- Technical explanation
Main Points Raised
- One participant states that $\tan\left({\arcsin\left({\frac{2}{3}}\right)}\right)=\frac{2\sqrt{5}}{5}$ but expresses confusion about this result.
- Another participant explains that $\arcsin\left(\frac{2}{3}\right)$ corresponds to an angle in a right triangle with sides in the ratio of 2:3, leading to the calculation of the adjacent side as $\sqrt{5}$ and thus confirming that $\tan\left(\arcsin\left(\frac{2}{3}\right)\right)=\frac{2}{\sqrt{5}}=\frac{2\sqrt{5}}{5}$.
- A different participant reiterates the confusion regarding the calculation of $\tan\left({\arcsin\left({\frac{2}{3}}\right)}\right)$ and suggests using the identity $\cos\left({\sin^{-1}\left({x}\right)}\right)=\sqrt{1-{x}^{2}}$ to find the cosine value.
- One participant questions whether $\cos(x)$ would cancel out when using the suggested identity.
- Another participant responds to the cancellation question, clarifying that the tangent formula does not lead to cancellation and provides a reformulation of the tangent in terms of sine and cosine.
Areas of Agreement / Disagreement
Participants express confusion and seek clarification on the calculations, indicating a lack of consensus on the understanding of the relationships between the trigonometric functions involved. Multiple viewpoints on the approach to the problem are present, and the discussion remains unresolved.
Contextual Notes
Participants rely on specific trigonometric identities and the properties of inverse functions, but there are indications of uncertainty regarding the application of these identities and the calculations involved.