Question

For what value of C will y = sin1/2(x - C) be an even function?

a. 2pi
b. pi
c. pi/2

Answer 1

c. pi/2

Explanation:

The answer is the option c. pi/2.

You must know that y = sin(x) is an odd function and also that y = cos(x) is an even function.

Also, you should know that sin(x + pi/2) = cos(x).

You can show it using the definition of the functions sine and cosine in the unit circle or using the formula of the sine of a sum: sin(A + B) = sin(A)*cos(B) + cos(A)*sin(B).

When you substitute B with pi/2 you get sin (A + pi/2) = sin(A)*0 + cos(A)*1 = cos(A).

Then, given that cos(A) is even sin(A+pi/2) is even.

Answer 2

option b

Explanation:

We are given that
`y=sin (1)/(2)(x-C)` be an even function

We have to find the value of C for which given function is even function

We know that sin x is odd function and cos is even function

Odd function : when f(x)
`\\eqf(-x)` then the function is called an odd function.

Even function : When f(x)=f(-x) then the function is called an even function.

Sin(-x)=-Sin x

Cos (-x)= Cos x

When we take C=
`2\pi`

Then , y=Sin
`(x)/(2)-(2\pi)/(2)`

y=
`sin((x)/(2)-\pi)`

`y=-sin(x)/(2)` (
`sin (x-\pi)=-sin x`)

When x is replace by -x

Then, we get
`y=-sin(-(x)/(2))=sin(x)/(2)`

`f(-x)\\eq f( x)`

Hence, option a is false.

b.C=
`\pi`

`y= sin ((x)/(2)-(\pi)/(2))`

`y=-sin((\pi)/(2)-(x)/(2))`

`y=-cos (x)/(2)`

When x is replaced by -x then we get

`y=-cos (-(x)/(2))=- cos (x)/(2)`

f(x)=f(-x) , Therefore, function is even,hence option b is true.

c.C=
`(\pi)/(2)`

`y=sin ((x)/(2)-(\pi)/(4))`

`Sin (A-B)=Sin A Cos B- Sin B Cos A`

`[y= sin (x)/(2) cos {(\pi)/(4)-cos(x)/(2) sin(\pi)/(4)`

`sin(\pi)/(4)= cos (\pi)/(4)=(1)/(\sqrt2)`

`y=(1)/(\sqrt2)(sin (x)/(2)- cos (x)/(2))`

When x is replaced by -x then we get

`y=(1)/(\sqrt2)(-sin(x)/(2)-cos (x)/(2))`

`f(x)\\eq f(-x)`

Hence, function is odd .Therefore, option c is false.