This identity means that translating a basic sine graph 3π/2 units to the right produces a basic cosine graph.
True or False?
sin(X-(3(pi)/2)) = cosX
let's test it by applying the sin(A - B) = sinAcosB - cosAsinB relation
sin(x - 3π/2)
= sinx cos3π/2 - cosx sin3π/2
= (sinx) (0) - (cosx)(-1)
= cosx
yup, it is true.
To determine whether the statement is true or false, we need to compare the graphs of the sine function and the cosine function and analyze how they are related.
The sine function and the cosine function are related by a phase shift of π/2 radians (or 90 degrees). When a sine function is shifted π/2 units to the right, it becomes a cosine function, and vice versa.
To verify this relationship mathematically, let's compare the given equation:
sin(X - (3π/2)) = cos(X)
Notice that the given equation subtracts 3π/2 units from the angle X in the sine function. If we simplify this equation, we can observe whether it matches the cosine function.
To simplify the equation, we distribute the negative sign:
sin(X)cos(3π/2) - cos(X)sin(3π/2) = cos(X)
Now, let's evaluate the trigonometric values of sine and cosine at 3π/2:
sin(3π/2) = -1
cos(3π/2) = 0
Substituting these values back into the equation:
sin(X)(0) - cos(X)(-1) = cos(X)
0 + cos(X) = cos(X)
As we can see, the equation simplifies to cos(X) = cos(X), which is true for all values of X. This confirms that the given equation is correct:
sin(X - (3π/2)) = cos(X)
Therefore, the statement is true. When you translate a basic sine graph 3π/2 units to the right, it produces a basic cosine graph.