How is cos(-x) = cos(x) ?
Trigonometric ratios deal with the relation between the angles and sides of a triangle.
Answer: cos(-x) = cos(x)
Using the complimentary angle properties of sine and cosine functions, let's prove it.
Cosine and Sine values are complimentary.
Thus, cos a = sin(90° - a)
⇒ cos(-x) = sin(90°+x)
We know that, sin(A + B) = sin A cos B + cos A sin B
Thus, using this formula we can expand sin(90° + x)
= sin 90° × cosx + cos90° × sin x
= 1 × cosx + 0 × sin x (Since, sin 90° = 1, cos 90° = 0)
= cos x
Cosine function is an even function that is mirrored perfectly around the y-axis.
So, for every absolute value on the x-axis, the value of y will be the same - whether the point x is chosen on the positive x-axis or the negative x-axis.
Thus, cos(-x) = cos(x)