QR problemsolving help

[Shakibanov]

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Someone help me work out this problem:
Sin(3x-26) =Cos(5x-60)

 

[Cher*]

You should solve this problem by converting either the sin function to cos or cos function to sin:

Use either of the following identities:

sin (x) = cos (pi/2 - x)
cos (x) = sin (pi/2 - x)

write out the equation as follows:

sin (3x-26) = cos (90-(3x-26)) = cos (116-3x)

since sin (3x-26) = cos (5x-60) :

cos (116-3x) = cos (5x-60)

according to the above equality, the following MUST hold true:

116-3x = 5x - 60

Solve for x:

8x = 176
x = 22


You can check the answer by plugging it back in the original formula. It works! 😉

 

[Streetwolf]

There's actually more, though I'm sure that's the answer they would be looking for.

If sin x = cos y then x + y = 90 + 360n where n = {all integers}

So we take 3x - 26 + 5x - 60 = 90 + 360n

8x - 86 = 90 + 360n

8x = 176 + 360n

x = 22 + 45n ; n = {all integers}

And obviously with n = 0 you get x = 22 which is the first answer. For n < 0 you get the negative angles and for n > 0 the positive ones. Plug those in too!

For example, n = 1 corresponds to x = 67, n = -1 corresponds to x = -23, etc.