Write the expression in terms of sine only.
8(sin 2x − cos 2x)
8(sin2x - cos2x)
8(sin2x - (1-2sin^2(x))
8(sin2x + 2sin^2(x) - 1)
To write the expression 8(sin 2x − cos 2x) in terms of sine only, we can make use of the trigonometric identity:
sin(x + y) = sin(x)cos(y) + cos(x)sin(y)
Let's rewrite the expression using this identity:
8(sin 2x − cos 2x) = 8(sin(2x)cos(0) + cos(2x)sin(0))
Since cos(0) = 1 and sin(0) = 0, the expression simplifies to:
8(sin(2x) * 1 + cos(2x) * 0)
Which further simplifies to:
8(sin(2x) * 1) = 8(sin 2x)
Therefore, the expression 8(sin 2x − cos 2x) in terms of sine only is simply 8sin(2x).
To write the given expression in terms of sine only, we can use the trigonometric identity:
sin(2x) = 2sin(x)cos(x)
Let's rewrite the expression:
8(sin 2x − cos 2x) = 8(2sin(x)cos(x) - cos(2x))
Now we will use the double angle identity for cosine:
cos(2x) = 1 - 2sin^2(x)
Substituting this back into the expression, we have:
8(2sin(x)cos(x) - (1 - 2sin^2(x)))
Expanding the expression, we get:
16sin(x)cos(x) - 8 + 16sin^3(x)
Finally, we can rewrite cos(x) in terms of sin(x):
cos(x) = √(1 - sin^2(x))
Substituting this back into the expression, we have:
16sin(x) * √(1 - sin^2(x)) - 8 + 16sin^3(x)
So, the given expression in terms of sine only is:
16sin(x) * √(1 - sin^2(x)) - 8 + 16sin^3(x)