Master inverse trigonometric functions (arcsin, arccos, arctan) and hyperbolic functions (sinh, cosh, tanh). Learn their derivatives, integrals, identities, and real-world applications.
What's the derivative of this inverse trigonometric function?
Find the derivative:
Calculate the value of this hyperbolic function!
Given:
Use: sinh(x) = (eĖ£ - eā»Ė£)/2
Inverse trigonometric functions reverse the action of trigonometric functions. They require restricted domains to ensure each input has a unique output (one-to-one correspondence).
Definition: arcsin(x) is the angle whose sine is x
Domain: [-1, 1] (range of sin(x))
Range: [-Ļ/2, Ļ/2] (principal values)
Key property: sin(arcsin(x)) = x for x ā [-1, 1]
Example: arcsin(1/2) = Ļ/6 because sin(Ļ/6) = 1/2
Definition: arccos(x) is the angle whose cosine is x
Domain: [-1, 1]
Range: [0, Ļ] (different from arcsin to ensure uniqueness)
Key property: cos(arccos(x)) = x for x ā [-1, 1]
Example: arccos(ā3/2) = Ļ/6 because cos(Ļ/6) = ā3/2
Definition: arctan(x) is the angle whose tangent is x
Domain: (-ā, ā) (all real numbers)
Range: (-Ļ/2, Ļ/2)
Key property: tan(arctan(x)) = x for all x
Example: arctan(1) = Ļ/4 because tan(Ļ/4) = 1