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ehrenfest
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Homework Statement
This always seemed intuitive to me, but when I tried to prove it I got stuck:
sin(x +pi/2) = cos(x)
It is easy with the angle addition formula, but is there another way?
ehrenfest said:Homework Statement
This always seemed intuitive to me, but when I tried to prove it I got stuck:
sin(x +pi/2) = cos(x)
It is easy with the angle addition formula, but is there another way?
Yes, this identity can be proven using the sum formula for sine and cosine. By substituting pi/2 for the angle in the sum formula, we can simplify the expression and show that sin(x + pi/2) = cos(x).
The intuitive explanation is that when x + pi/2 is added to an angle, it results in a 90 degree rotation of the original angle. This rotation changes the sine value to the cosine value, hence the identity sin(x + pi/2) = cos(x).
The unit circle is a circle with a radius of 1 centered at the origin on a coordinate plane. The x-axis represents the cosine values and the y-axis represents the sine values. When an angle is rotated by pi/2, it moves clockwise or counterclockwise along the unit circle, changing the sine value to the cosine value and vice versa.
No, this identity is true for all values of x. It is a fundamental trigonometric identity that holds true for any angle in radians or degrees.
From this identity, we can also derive the identities cos(x + pi/2) = -sin(x) and sin(x - pi/2) = -cos(x). These identities can be useful when solving trigonometric equations and simplifying expressions involving sine and cosine.