cos(54º-X)=sin2X
To find the value of X that satisfies the equation cos(54º-X) = sin(2X), we can use trigonometric identities and algebraic manipulation.
First, let's rewrite the equation using the double angle formulas for sine and cosine:
cos(54º-X) = 2sin(X)cos(X)
Next, let's apply the cosine addition formula:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
cos(54º-X) = cos(54º)cos(X) + sin(54º)sin(X)
Now, we can compare the terms on both sides of the equation and set them equal to each other:
cos(54º)cos(X) + sin(54º)sin(X) = 2sin(X)cos(X)
Next, we'll rearrange the equation by moving all terms to one side:
cos(54º)cos(X) + sin(54º)sin(X) - 2sin(X)cos(X) = 0
Now, let's simplify the terms:
cos(54º)cos(X) - sin(54º)sin(X) - 2sin(X)cos(X) = 0
Using the trigonometric identity cos(A - B) = cos(A)cos(B) + sin(A)sin(B), we can rewrite the left side of the equation:
cos(54º - X) - 2sin(X)cos(X) = 0
Now, we can substitute back the original equation:
cos(54º - X) - sin(2X) = 0
Finally, we can solve this equation using algebraic methods or numerical methods to find the value(s) of X that satisfy the equation.