Part 1: Using complete sentences, compare the key features and graphs of sine and cosine. What are their similarities and differences?
Answer: The sine graph will always pass through (0, 0), While the cosine graph wouldn't. If the graph is 2cosx, it will pass through (0, 2). How high the graphs go depend on its amplitude, (how high or how far the graphs go can sometimes be infinite).
Part 2: Using these similarities and differences, how would you transform f(x) = 3 sin(4x - π) + 4 into a cosine function in the form f(x) = a cos(bx - c) + d?
Answer: I don't know, please help
trig identity:
sin theta = cos (pi/2-theta)
so
y = 3 sin (4x-pi) + 4
but from above trig identity
y = 3 cos [ pi/2 - (4x-pi) ] + 4
or
y = 3 cos (-4x + 3 pi/2) + 4
but
cos (- theta) = cos theta
so
y = 3 cos (4x-3 pi/2) + 4
Thank you
also in your explanation explain that the cosine curve is 90 degrees or pi/2 radians behind the sine curve if you graph y = sin theta and y = cos theta versus theta
You are welcome :)
By the way, there are an infinite number of solutions. eg 3 cos (4x - 7 pi/2)+4
or
- 3 cos(4x -pi/2) + 4
etc
3-x/(x+3)(x-3)
is this right?
this helped me a lot
To transform the given function, f(x) = 3 sin(4x - π) + 4, into a cosine function in the form f(x) = a cos(bx - c) + d, you can use the similarities and differences between sine and cosine.
1. Start by identifying the similarities and differences between sine and cosine functions.
- Both sine and cosine functions are periodic with a period of 2π.
- Sine is an odd function, while cosine is an even function.
- Sine passes through (0, 0), while cosine does not.
2. Transforming the given function into a cosine function:
- Since sine is an odd function and cosine is an even function, we can use the fact that sin(x) = cos(x - π/2) to transform the equation.
- Substitute sin(4x - π) with cos(4x - π - π/2) in the given function to get f(x) = 3 cos(4x - 3π/2) + 4.
- Here, a = 3, b = 4, c = 3π/2, and d = 4.
Therefore, the transformed cosine function is f(x) = 3 cos(4x - 3π/2) + 4.