Answer :
To find the expression for the given function [tex]\( f(x) = \frac{3}{4} \cos \left(2x + 90^\circ \right) \)[/tex], we can follow these steps:
### Step-by-Step Process:
1. Understand the angle units:
The [tex]\( 90^\circ \)[/tex] is in degrees, and we will have to convert all angle measurements to radians because the cosine function in most standard mathematical settings, and especially in higher calculus, uses radians.
2. Convert degrees to radians:
Recall that [tex]\( 90^\circ \)[/tex] is equivalent to [tex]\( \frac{\pi}{2} \)[/tex] radians. So, the function transforms into:
[tex]\[ f(x) = \frac{3}{4} \cos \left(2x + \frac{\pi}{2} \right) \][/tex]
3. Use the trigonometric identity for cosine:
One of the trigonometric identities for cosine is:
[tex]\[ \cos \left(\theta + \frac{\pi}{2} \right) = -\sin(\theta) \][/tex]
Hence, substituting [tex]\( \theta = 2x \)[/tex]:
[tex]\[ \cos \left(2x + \frac{\pi}{2} \right) = -\sin(2x) \][/tex]
4. Substitute back into the function:
Using this identity, the function [tex]\( f(x) \)[/tex] becomes:
[tex]\[ f(x) = \frac{3}{4} \cos \left(2x + \frac{\pi}{2} \right) = \frac{3}{4} \left( -\sin(2x) \right) \][/tex]
5. Simplify the expression:
Therefore, the expression simplifies to:
[tex]\[ f(x) = -\frac{3}{4} \sin(2x) \][/tex]
### Final Result:
[tex]\[ f(x) = -\frac{3}{4} \sin(2x) \][/tex]
This is the simplified form of the given function [tex]\( f(x) \)[/tex].
### Step-by-Step Process:
1. Understand the angle units:
The [tex]\( 90^\circ \)[/tex] is in degrees, and we will have to convert all angle measurements to radians because the cosine function in most standard mathematical settings, and especially in higher calculus, uses radians.
2. Convert degrees to radians:
Recall that [tex]\( 90^\circ \)[/tex] is equivalent to [tex]\( \frac{\pi}{2} \)[/tex] radians. So, the function transforms into:
[tex]\[ f(x) = \frac{3}{4} \cos \left(2x + \frac{\pi}{2} \right) \][/tex]
3. Use the trigonometric identity for cosine:
One of the trigonometric identities for cosine is:
[tex]\[ \cos \left(\theta + \frac{\pi}{2} \right) = -\sin(\theta) \][/tex]
Hence, substituting [tex]\( \theta = 2x \)[/tex]:
[tex]\[ \cos \left(2x + \frac{\pi}{2} \right) = -\sin(2x) \][/tex]
4. Substitute back into the function:
Using this identity, the function [tex]\( f(x) \)[/tex] becomes:
[tex]\[ f(x) = \frac{3}{4} \cos \left(2x + \frac{\pi}{2} \right) = \frac{3}{4} \left( -\sin(2x) \right) \][/tex]
5. Simplify the expression:
Therefore, the expression simplifies to:
[tex]\[ f(x) = -\frac{3}{4} \sin(2x) \][/tex]
### Final Result:
[tex]\[ f(x) = -\frac{3}{4} \sin(2x) \][/tex]
This is the simplified form of the given function [tex]\( f(x) \)[/tex].
