Draw a triangle in the plane with vertices (1,x) ( 1, x), (1,0) ( 1, 0), and the origin. Then arctan(x) arctan ( x) is the angle between the positive x-axis and the ray beginning at the origin and
sin(arccos(x)) = √1 −x2 Explanation: From Pythagoras, we have: sin2θ+ cos2θ = 1 If x ∈ [ − 1,1] and θ = arccos(x) then: θ ∈ [0,π] sin(θ) ≥ 0 Hence: sin(arccos(x)) = sin(θ) = √1
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Draw a triangle in the plane with vertices (1,x) ( 1, x), (1,0) ( 1, 0), and the origin. Then arctan(x) arctan ( x) is the angle between the positive x-axis and the ray beginning at the origin and
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arctan (x) = x 1 Considering the sides of the right triangle. We have o p p o s i t e s i d e = x, a d j a c e n t s i d e = 1 and h y p o t e n u s e = x 2 + 1 Therefore the sine of this angle =
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