Solve the equation sin 2@=cos 3@

Answer 1

To solve the equation sin(2@) = cos(3@), we can use trigonometric identities to simplify the equation.

Recall the double-angle identity for sine: sin(2@) = 2sin@cos@

Also recall the angle addition formula for cosine: cos(3@) = cos(2@ + @) = cos(2@)cos(@) - sin(2@)sin(@)

Now substitute these identities into the original equation:

2sin@cos@ = cos(2@)cos(@) - sin(2@)sin(@)

Next, use the double-angle identity for cosine: cos(2@) = 2cos^2@ - 1

2sin@cos@ = (2cos^2@ - 1)cos(@) - sin(2@)sin(@)

Now we need to expand and simplify the equation:

2sin@cos@ = 2cos^3@ - cos(@) - 2sin^2@cos@

2sin@cos@ = 2cos^3@ - cos(@) - 2(1 - cos^2@)cos@

2sin@cos@ = 2cos^3@ - cos(@) - 2cos@ + 2cos^3@

2sin@cos@ = 4cos^3@ - 3cos@

Now, divide through by cos@ to solve for sin@:

2sin@ = 4cos^2@ - 3

sin@ = (4cos^2@ - 3) / 2

This is the solution to the equation sin(2@) = cos(3@)