Respuesta :

Heather

Answer: True.

Step-by-step explanation:

    [tex]\displaystyle cos(\frac{\pi}{2}-x)=sin(x)[/tex] is true. This equation is proven true by the co-function trigonometric identity. The co-function identities are:

         [tex]\displaystyle sin(\frac{\pi}{2}-x)=cos(x)[/tex]

         [tex]\displaystyle cos(\frac{\pi}{2}-x)=sin(x)[/tex]

         [tex]\displaystyle cot(\frac{\pi}{2}-x)=tan(x)[/tex]

         [tex]\displaystyle tan(\frac{\pi}{2}-x)=cot(x)[/tex]

         [tex]\displaystyle sec(\frac{\pi}{2}-x)=csc(x)[/tex]

         [tex]\displaystyle csc(\frac{\pi}{2}-x)=sec(x)[/tex]

Intriguing456

Answer:

True

Step-by-step explanation:

First, we can rewrite the input to the cosine function so that it resembles the form:

[tex]a\cos(b(x-c)) + d[/tex]

where the relevant variables are:

  • [tex]b[/tex] = horizontal stretch
          • period = [tex]\dfrac{2\pi}{b}[/tex]
  • [tex]c[/tex] = horizontal shift

↓↓↓

[tex]\cos\!\left(\dfrac{\pi}2 - x\right) = \cos\!\left(-\!\!\left(x-\dfrac{\pi}2\right)\right)[/tex]

So, we can identify the variable values as:

  • [tex]b = -1[/tex]
  • [tex]c = \pi/2[/tex]   (notice how [tex]c[/tex] is subtracted in the equation form)

Note that a negative b will not affect the cosine function because it is even (symmetrical about the y-axis). We know that for even functions:

[tex]f(-x)=f(x)[/tex]

Hence:

[tex]\cos(-x) = \cos(x)[/tex]

What will affect the cosine function is the horizontal shift. Since the sine and cosine functions are shifted versions of each other by a difference of π/2, we can say that the given statement is TRUE because we can identify that:

  • [tex]b = -\dfrac{\pi}{2}[/tex]

from the given cosine function:

[tex]\cos\!\left(-\!\!\left(x-\dfrac{\pi}2\right)\right)[/tex]

Ver imagen Intriguing456

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