If sin(2x) = cos (x + 30%), what is the value of x?!

  x ∈ {20°, 120°, 140°, 260°}

Step-by-step explanation:

I find a graphing calculator useful for solving equations of this sort. Subtracting cos(x+30°) from both sides gives an expression that is zero when x has the appropriate value. The calculator is very good at finding zeros.

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We can make use of the formula for the sum of two sines.

  sin(a) +sin(b) = 2sin((a+b)/2)cos((a-b)/2)

Then we can rewrite ...

  sin(2x) -sin(x +120°) = 0

as ...

  2sin((2x-(x+120°))/2)·cos((2x+(x+120°))/2) = 0

  sin((x -120°)/2)·cos((3x +120°)/2) = 0 . . . . . divide by 2 and simplify a bit

This resolves to two equations according to the zero product rule.

  x/2 -60° = k·180° . . . . . for some integer k

  x = 120° +k·360° . . . . . for some integer k

And the other equation is ...

  3x/2 +60° = (2k +1)·90° . . . . . for some integer k

  3x/2 = k·180° +30° . . . . . . . . subtract 60°, eliminate parentheses

  3x = k·360° +60°

  x = k·120° +20° . . . . . for some integer k

The values shown in the Answer section above are the values of x in the range 0 to 360°: 20°, 120°, 140°, 260°.

irrational, then, just means all the numbers that aren't rational. every integer is a rational number, since each integer n can be written in the form n/1. however, numbers like 1/2, 45454737/2424242, and -3/7 are also rational, since they are fractions whose numerator and denominator are integers.

step-by-step explanation: