Answer:
y = 2 sin (x + [tex]\frac{3\pi }{4}[/tex] ) has the same graph of y = 2 cos (x + [tex]\frac{\pi }{4}[/tex] ) ⇒ 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 + [tex]\frac{\pi }{4}[/tex] )
∵ 2 cos (x + [tex]\frac{\pi }{4}[/tex] ) = 2[cos x cos [tex]\frac{\pi }{4}[/tex] - sin x sin [tex]\frac{\pi }{4}[/tex] ]
∵ cos [tex]\frac{\pi }{4}[/tex] = [tex]\frac{\sqrt{2}}{2}[/tex] and sin [tex]\frac{\pi }{4}[/tex] = [tex]\frac{\sqrt{2}}{2}[/tex]
- Substitute them in the right hand side
∴ 2 cos (x + [tex]\frac{\pi }{4}[/tex] ) = 2[ [tex]\frac{\sqrt{2}}{2}[/tex] cos x - [tex]\frac{\sqrt{2}}{2}[/tex] sin x]
- Multiply the bracket in the right hand side by 2
∴ 2 cos (x + [tex]\frac{\pi }{4}[/tex] ) = [tex]\sqrt{2}[/tex] cos x - [tex]\sqrt{2}[/tex] sin x
∴ y = [tex]\sqrt{2}[/tex] cos x - [tex]\sqrt{2}[/tex] sin x
Now let us find the function which give the same right hand side of the function above
y = 2 sin (x + [tex]\frac{3\pi }{4}[/tex] )
∵ 2 sin (x + [tex]\frac{3\pi }{4}[/tex] ) = 2[sin x cos [tex]\frac{3\pi }{4}[/tex] + cos x sin [tex]\frac{3\pi }{4}[/tex] ]
∵ sin [tex]\frac{3\pi }{4}[/tex] = [tex]\frac{\sqrt{2}}{2}[/tex] and cos [tex]\frac{3\pi }{4}[/tex] = [tex]-\frac{\sqrt{2}}{2}[/tex]
- Substitute them in the right hand side
∴ 2 sin (x + [tex]\frac{3\pi }{4}[/tex] ) = 2[ [tex]-\frac{\sqrt{2}}{2}[/tex] sin x + [tex]\frac{\sqrt{2}}{2}[/tex] cos x]
- Multiply the bracket in the right hand side by 2
∴ 2 sin (x + [tex]\frac{3\pi }{4}[/tex] ) = [tex]-\sqrt{2}[/tex] sin x + [tex]\sqrt{2}[/tex] cos x
- Switch the two terms of the right hand side
∴ 2 sin (x + [tex]\frac{3\pi }{4}[/tex] ) = [tex]\sqrt{2}[/tex] cos x - [tex]\sqrt{2}[/tex] sin x
∴ y = [tex]\sqrt{2}[/tex] cos x - [tex]\sqrt{2}[/tex] sin x
- The same with right hand side of the function above
∴ y = 2 sin (x + [tex]\frac{3\pi }{4}[/tex] ) has the same graph of y = 2 cos (x + [tex]\frac{\pi }{4}[/tex] )
