Find the value of x

Question

Sin(5x-28)°=cos(3x-50) °

Answer 1

The cosine of an angle is the same as the sine of its complement.

sin(5x-28)=sin(90-(3x-50)), 5x-28=90-3x+50, 8x=90+50+28=168, so x=21º. This is one of many answers.

The sine of an angle is also the sine of its supplement:

180-5x+28=90-3x+50, 180-90-50+28=2x, 68=2x, x=34º.

Other answers are 66º, 111º, etc, because of the cyclic nature of the harmonic functions.

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To find all the angles we modify the above equations.

(1) 8x=168+360n, x=21+45n where n is any integer, positive or negative or zero. So we get the angles 21, 66, 111, 156, ... degrees from this formula. Also, -24, -69, ... degrees.

(2) The sine is positive in the first and second quadrant so the other formula is 2x=68+360n, x=34+180n and x=34, 214, 394, ..., -146, -326, ... degrees.

There are an infinite number of solutions.