I do not know how to find a single function of x or ø by using an identity. How do you find a single function using identities for a problem such as:
cos(3pi/4 - x)
huh? This is a single function!
Not sure what you're getting at...
y = cos(3π/4 - x) already is a single function of x, namely the cosine function.
I don't understand what else you want to do with it.
We know:
using cos(A-B) = cosAcosB + sinAsinB
cos(3π/4 - x) = cos 3π/4 cosx + sin 3π/4 sin x
= -√2/2 cosx + √2/2sinx
= √2/2 (sinx - cosx) , no improvement here.
To find a single function of x using identities, you need to use trigonometric identities to simplify the given expression. In this case, we are given the expression cos(3π/4 - x).
Let's start by using the cosine difference identity, which states that cos(A - B) = cos(A)cos(B) + sin(A)sin(B). Applying this identity to our expression, we have:
cos(3π/4 - x) = cos(3π/4)cos(x) + sin(3π/4)sin(x)
Now, we need to determine the values of cos(3π/4) and sin(3π/4). Since π/4 is a commonly known angle, we can determine the values using the unit circle. For an angle of π/4, we know that cos(π/4) = sin(π/4) = 1/√2.
In the unit circle, 3π/4 is located in the second quadrant, where the x-coordinate is negative and the y-coordinate is positive. Therefore, in this quadrant, cos(3π/4) = -1/√2 and sin(3π/4) = 1/√2.
Substituting these values into our expression, we get:
cos(3π/4 - x) = (-1/√2)cos(x) + (1/√2)sin(x)
Now, this expression represents a single function of x.
In summary, to find a single function of x using trigonometric identities, you would start by applying the relevant identities to simplify the expression. Then, if there are any unknown trigonometric values, you would determine them using the unit circle or other known methods. Finally, you would substitute the determined values into the expression to obtain a single function of x.