Question

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

Answer 1

Answer:

y = 2 sin (x + $\frac{3\pi }{4}$ ) has the same graph of y = 2 cos (x + $\frac{\pi }{4}$ ) ⇒ 3rd answer

Step-by-step 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 + $\frac{\pi }{4}$ )

∵ 2 cos (x + $\frac{\pi }{4}$ ) = 2[cos x cos $\frac{\pi }{4}$  - sin x sin $\frac{\pi }{4}$ ]

∵ cos $\frac{\pi }{4}$  = $\frac{\sqrt{2}}{2}$  and sin $\frac{\pi }{4}$  = $\frac{\sqrt{2}}{2}$  

- Substitute them in the right hand side

∴ 2 cos (x + $\frac{\pi }{4}$ ) = 2[ $\frac{\sqrt{2}}{2}$ cos x  -  $\frac{\sqrt{2}}{2}$ sin x]

- Multiply the bracket in the right hand side by 2

∴ 2 cos (x + $\frac{\pi }{4}$ ) = $\sqrt{2}$ cos x - $\sqrt{2}$ sin x

∴ y = $\sqrt{2}$ cos x - $\sqrt{2}$ sin x

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

y = 2 sin (x + $\frac{3\pi }{4}$ )

∵ 2 sin (x + $\frac{3\pi }{4}$ ) = 2[sin x cos $\frac{3\pi }{4}$  + cos x sin $\frac{3\pi }{4}$ ]

∵ sin $\frac{3\pi }{4}$  = $\frac{\sqrt{2}}{2}$  and cos $\frac{3\pi }{4}$  = $-\frac{\sqrt{2}}{2}$  

- Substitute them in the right hand side

∴ 2 sin (x + $\frac{3\pi }{4}$ ) = 2[ $-\frac{\sqrt{2}}{2}$ sin x  + $\frac{\sqrt{2}}{2}$ cos x]

- Multiply the bracket in the right hand side by 2

∴ 2 sin (x + $\frac{3\pi }{4}$ ) =  $-\sqrt{2}$ sin x  + $\sqrt{2}$ cos x

- Switch the two terms of the right hand side

∴ 2 sin (x + $\frac{3\pi }{4}$ ) =  $\sqrt{2}$ cos x - $\sqrt{2}$ sin x  

∴ y =  $\sqrt{2}$ cos x - $\sqrt{2}$ sin x

- The same with right hand side of the function above

∴ y = 2 sin (x + $\frac{3\pi }{4}$ ) has the same graph of y = 2 cos (x + $\frac{\pi }{4}$ )