What is the general solution for sin(x)-cos(x)= 1/3 ,sin(2x) ?

The general solution for sin(x)-cos(x)= 1/3 ,sin(2x) is No Solution for x\in\mathbb{R}

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sin(x)-cos(x)= 1/3 ,sin(2x)

sin (x) - cos (x) = \frac{1}{3}, sin (2 x)

No Solution for x \in R

Solution steps

sin (x) - cos (x) = \frac{1}{3}, sin (2 x)

Rewrite using trig identities

2 sin (x - \frac{π}{4}) = \frac{1}{3}

Divide both sides by

2

sin (x - \frac{π}{4}) = \frac{2}{6}

Apply trig inverse properties

x - \frac{π}{4} = arcsin (\frac{2}{6}) + 2 πn, x - \frac{π}{4} = π - arcsin (\frac{2}{6}) + 2 πn

Solve

x - \frac{π}{4} = arcsin (\frac{2}{6}) + 2 πn : x = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Solve

x - \frac{π}{4} = π - arcsin (\frac{2}{6}) + 2 πn : x = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

x = arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}, x = π - arcsin (\frac{1}{3 2}) + 2 πn + \frac{π}{4}

Solutions for the range

sin (2 x)

No Solution for x \in R

- 9 cos (θ) = \frac{3 2}{2}

sec (a) = 1 + tan (a)

arctan (θ) = \frac{2}{2 3}

3 sin (x) = + sin (x)

x, (D^{2} - 3 D + 2) y = sec^{2} (e^{- x})

What is the general solution for sin(x)-cos(x)= 1/3 ,sin(2x) ?
The general solution for sin(x)-cos(x)= 1/3 ,sin(2x) is No Solution for x\in\mathbb{R}