find cos(π/2+x)
fast no wrong answers
Answers
Answer:
1.57+x
(What does the variable Equal?)
Answer:
How can I simply prove that cos(π/2-x)=sinx?
Sachidanand Das
Updated November 7, 2018
Proof 1:
The simplest way to prove
cos(π/2 - x) = sin x
is to put A = π/2 , B = x in the trigonometric formula
cos(A-B) = cos A . cos B + sin A . sin B……………………………….(1)
and obtain
cos(π/2 - x) = cos π/2 . cos x + sin π/2 . sin x……………………….(2)
Substituting cos π/2 = 0 and sin π/2 = 1 in (2),
cos (π/2 - x) = 0 . cos x + 1 . sin x=0+sin x
∴cos (π/2 - x) = sin x (Proved)
Proof 2:
Let ABC be a triangle right-angled at B. Let AB be the base and AC the hypotenuse. If we denote the angle C by x, the base angle A = (π/2 - x) so that A + B + C = π/2 - x + π/2 + x =π or 180° .
Now for the base angle A, BC is the perpendicular.
∴ cos A = cos (π/2 - x) = base/hypotenuse = AB/AC …………..(3)
For the angle C, AB is the perpendicular and therefore
sin C = sin x = perpendicular/hypotenuse = AB/AC…………….(4)
Equating (3) and (4),
cos (π/2 - x) = sin x (Proved)
Proof 3:
Use the Euler’s formula
eⁱᶿ = cos θ + i sin θ
which defines the symbol eⁱᶿ for any real value of θ . Here i = √-1 .
∴ We can put θ = (π/2 - x) in the formula and write
e^i(π/2 - x) = cos (π/2 - x) + i sin (π/2 - x)
Or, e^iπ/2 . e^(-ix) = cos (π/2 - x) + i sin (π/2 - x)
Now e^iπ/2 = cos π/2 + i sin π/2 = 0 + i.1 = i and e^(-ix) = cos x - i sinx
∴i .( cos x - i sin x) = cos (π/2 - x) + i sin (π/2 - x)
Or, i cos x + sin x = cos (π/2 - x) + i sin (π/2 - x) [Since i² =-1]
Equating the real and imaginary parts,
cos (π/2 - x) = sin x (Proved)
and cos x = sin (π/2 - x)
Concluding Remarks:
Of the three methods presented here for proving the given assertion , the preferred method should be the Proof 1. This is because it is simple, straight forward and fast. Can be done mentally by an average student in about 30 seconds. In Proof 2, there is room for confusion as to which is the base, which is the right perpendicular to be taken. Besides one needs to spend extra time to draw a triangle, mark the sides, the angles, etc. Proof 3 is fine; but not many are comfortable or good at working with complex functions. The method entails more algebra than the other methods; but it gives a bonus, namely: it proves the formula cos x = sin (π/2 - x).