SUMMARY
The discussion focuses on calculating $\cos\left({\sin^{-1}\left({\frac{2}{3}}\right)}\right)$ using the identity $\cos\left({\sin^{-1}\left({x}\right)}\right)=\sqrt{1-{x}^{2}}$. Participants confirm that for $x = \frac{2}{3}$, this results in $\cos\left({\sin^{-1}\left({\frac{2}{3}}\right)}\right) = \sqrt{1 - \left(\frac{2}{3}\right)^2} = \sqrt{\frac{5}{9}} = \frac{\sqrt{5}}{3}$. Additionally, they derive the tangent of the angle using the ratio of the opposite side (2) to the hypotenuse (3), resulting in $\tan\left({\arcsin\left({\frac{2}{3}}\right)}\right) = \frac{2\sqrt{5}}{5}$.
PREREQUISITES
- Understanding of trigonometric identities, specifically $\cos\left({\sin^{-1}\left({x}\right)}\right)$
- Knowledge of the Pythagorean theorem
- Familiarity with the concepts of sine, cosine, and tangent functions
- Ability to manipulate square roots and fractions in mathematical expressions
NEXT STEPS
- Study the derivation and applications of the identity $\cos\left({\sin^{-1}\left({x}\right)}\right)=\sqrt{1-{x}^{2}}$
- Learn how to calculate $\tan\left({\arcsin\left({x}\right)}\right)$ using right triangle definitions
- Explore the relationship between sine, cosine, and tangent in various quadrants
- Practice solving similar problems involving inverse trigonometric functions and their properties
USEFUL FOR
Students and educators in mathematics, particularly those studying trigonometry and inverse functions, as well as anyone seeking to deepen their understanding of trigonometric identities and calculations.