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

Answer:

c. pi/2

Step-by-step 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

Answer:

option b

Step-by-step explanation:

We are given that $y=sin \frac{1}{2}(x-C)$ 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)$\neqf(-x)$ 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$\frac{x}{2}-\frac{2\pi}{2}$

y=$sin(\frac{x}{2}-\pi)$

$y=-sin\frac{x}{2}$  ( $sin (x-\pi)=-sin x$)

When x is replace by -x

Then, we get $y=-sin(-\frac{x}{2})=sin\frac{x}{2}$

$f(-x)\neq f( x)$

Hence, option a is false.

b.C=$\pi$

$y= sin (\frac{x}{2}-\frac{\pi}{2})$

$y=-sin(\frac{\pi}{2}-\frac{x}{2})$

$y=-cos \frac{x}{2}$

When x is replaced by -x then we get

$y=-cos (-\frac{x}{2})=- cos \frac{x}{2}$

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

c.C=$\frac{\pi}{2}$

$y=sin (\frac{x}{2}-\frac{\pi}{4})$

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

$[y= sin \frac{x}{2} cos {\frac{\pi}{4}-cos\frac{x}{2} sin\frac{\pi}{4}$

$sin\frac{\pi}{4}= cos \frac{\pi}{4}=\frac{1}{\sqrt2}$

$y=\frac{1}{\sqrt2}(sin \frac{x}{2}- cos \frac{x}{2})$

When x is replaced by -x then we get

$y=\frac{1}{\sqrt2}(-sin\frac{x}{2}-cos \frac{x}{2})$

$f(x)\neq f(-x)$

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