Question

When is sinx=cos2x?

Answer 1

if you graph the two on a polar coordinate system, at the minimum value of the sin(x), and every other minimum value of cos(2x), they are the same values.

Answer 2

cos² x - sin² x = 1- 2 sin x cos x demostrate that this is a triginometric identity

Answer 3

From the compound angle formula of cosine and the trigonometric identity of sin²x + cos²x = 1, cos2x = cos²x－sin²x = (1－sin²x)－sin²x = 1－2sin²x.**    So, the given equation can be rewritten as follows: sinx = 1－sin²x ··· Eq.1

Let sinx = X, so the Eq.1 can be rewritten: X = 1－2X².    Put this into the standard form of quadratic equation: 2X + X－1 = 0.   To evaluate X, use the quadratic formula; x = (－b ± √(b²－4ac)) / 2a, where a = 2, b = 1 and c =－1.   So, X = (－1 ± √9) / 4, we have: X = 1/2 or X = -1.

Evaluate x using the unit circle.   When X = 1/2, sinx = 1/2.    So, x = (2n + 1/6)π or, (2n + 5/6)π.    And when X = -1, sinx = -1.   So, x = (2n + 3/2)π (n: any integers)

Therefore, when x=(2n+1/6)π or (2n+5/6)π, sinx=cos2x=1/2.   When x=(2n+3/2)π, sinx=cosx= -1

** ,or you can use the double-angle formula of cosine: cos2x = 1－2sin²x.