if I have sin(6x)+cos(6x)how would I write it in terms of sin only?
To write the expression sin(6x) + cos(6x) in terms of sin only, we can use a trigonometric identity known as the sine addition formula.
The sine addition formula states:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In our case, if we let a = 6x and b = π/4, we can rewrite the expression as follows:
sin(6x + π/4) = sin(6x)cos(π/4) + cos(6x)sin(π/4)
Now, let's recall the values of cos(π/4) and sin(π/4). The cosine of π/4 is 1/sqrt(2), and the sine of π/4 is also 1/sqrt(2).
Substituting these values back into the equation, we get:
sin(6x + π/4) = sin(6x) * (1/sqrt(2)) + cos(6x) * (1/sqrt(2))
Now, we can factor out 1/sqrt(2) from both terms:
sin(6x + π/4) = (1/sqrt(2)) * (sin(6x) + cos(6x))
Therefore, we have expressed sin(6x) + cos(6x) in terms of sin only as:
sin(6x + π/4) = (1/sqrt(2)) * (sin(6x) + cos(6x))