Question

Which function has the same graph as y = 2cos(x + pi/4 )?

y = 2sin(x + pi/4 )
y = 2sin(x - pi/4 )
y = 2sin(x + 3pi/4 )
y = 2sin(x - 3pi/4 )

Answer 1

y=2sin(x+3pi/4)

Explanation:

Answer C

Answer 2

y = 2 sin (x +
`(3\pi )/(4)` ) has the same graph of y = 2 cos (x +
`(\pi )/(4)` ) ⇒ 3rd answer

Explanation:

Let us revise the rules of the trigonometric compound angles

cos(x + y) = cos x cos y - sin x sin y

sin(x + y) = sin x cos y + cos x sin y

Let us solve the problem using the rules above

y = 2 cos (x +
`(\pi )/(4)` )

∵ 2 cos (x +
`(\pi )/(4)` ) = 2[cos x cos
`(\pi )/(4)` - sin x sin
`(\pi )/(4)` ]

∵ cos
`(\pi )/(4)` =
`(√(2))/(2)` and sin
`(\pi )/(4)` =
`(√(2))/(2)`

- Substitute them in the right hand side

∴ 2 cos (x +
`(\pi )/(4)` ) = 2[
`(√(2))/(2)` cos x -
`(√(2))/(2)` sin x]

- Multiply the bracket in the right hand side by 2

∴ 2 cos (x +
`(\pi )/(4)` ) =
`√(2)` cos x -
`√(2)` sin x

∴ y =
`√(2)` cos x -
`√(2)` sin x

Now let us find the function which give the same right hand side of the function above

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

∵ 2 sin (x +
`(3\pi )/(4)` ) = 2[sin x cos
`(3\pi )/(4)` + cos x sin
`(3\pi )/(4)` ]

∵ sin
`(3\pi )/(4)` =
`(√(2))/(2)` and cos
`(3\pi )/(4)` =
`-(√(2))/(2)`

- Substitute them in the right hand side

∴ 2 sin (x +
`(3\pi )/(4)` ) = 2[
`-(√(2))/(2)` sin x +
`(√(2))/(2)` cos x]

- Multiply the bracket in the right hand side by 2

∴ 2 sin (x +
`(3\pi )/(4)` ) =
`-√(2)` sin x +
`√(2)` cos x

- Switch the two terms of the right hand side

∴ 2 sin (x +
`(3\pi )/(4)` ) =
`√(2)` cos x -
`√(2)` sin x

∴ y =
`√(2)` cos x -
`√(2)` sin x

- The same with right hand side of the function above

∴ y = 2 sin (x +
`(3\pi )/(4)` ) has the same graph of y = 2 cos (x +
`(\pi )/(4)` )