Asked by Matthew
Write the following trigonometric expression as an algebraic expression in x free of trigonometric or inverse trigonometric functions
sin(cos^-1 x) -1≤x≤1
sin(cos^-1 x) -1≤x≤1
Answers
Answered by
Steve
draw a triangle with adjacent/hypotenuse = x/1
the other leg is √(1-x^2)
so, sin(arccos(x)) = √(1-x^2)
or, if θ = arccos(x), then we have the ubiquitous identity
sin^2θ + cos^2θ = 1
but cosθ = x, so
sin^2θ + x^2 = 1
sin^2θ = 1-x^2
sinθ = √(1-x^2)
the other leg is √(1-x^2)
so, sin(arccos(x)) = √(1-x^2)
or, if θ = arccos(x), then we have the ubiquitous identity
sin^2θ + cos^2θ = 1
but cosθ = x, so
sin^2θ + x^2 = 1
sin^2θ = 1-x^2
sinθ = √(1-x^2)