What is the general solution for sin(x)-cos(2x)-1=0 ?

The general solution for sin(x)-cos(2x)-1=0 is x=0.89590…+2pin,x=pi-0.89590…+2pin

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sin(x)-cos(2x)-1=0

Solution

sin (x) - cos (2 x) - 1 = 0

Solution

x = 0.89590 \dots + 2 πn, x = π - 0.89590 \dots + 2 πn

Degrees

x = 51.33171 \dots^{\circ} + 36 0^{\circ} n, x = 128.66828 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (x) - cos (2 x) - 1 = 0

Rewrite using trig identities

- 1 - cos (2 x) + sin (x)

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= - 1 - (1 - 2 sin^{2} (x)) + sin (x)

Simplify

- 1 - (1 - 2 sin^{2} (x)) + sin (x) : 2 sin^{2} (x) + sin (x) - 2

- 1 - (1 - 2 sin^{2} (x)) + sin (x)

- (1 - 2 sin^{2} (x)) : - 1 + 2 sin^{2} (x)

- (1 - 2 sin^{2} (x))

Distribute parentheses

= - (1) - (- 2 sin^{2} (x))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + 2 sin^{2} (x)

= - 1 - 1 + 2 sin^{2} (x) + sin (x)

Subtract the numbers:

- 1 - 1 = - 2

= 2 sin^{2} (x) + sin (x) - 2

= 2 sin^{2} (x) + sin (x) - 2

- 2 + sin (x) + 2 sin^{2} (x) = 0

Solve by substitution

- 2 + sin (x) + 2 sin^{2} (x) = 0

Let:

sin (x) = u

- 2 + u + 2 u^{2} = 0

- 2 + u + 2 u^{2} = 0 : u = \frac{- 1 + 17}{4}, u = \frac{- 1 - 17}{4}

- 2 + u + 2 u^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

2 u^{2} + u - 2 = 0

Solve with the quadratic formula

2 u^{2} + u - 2 = 0

Quadratic Equation Formula:

For

a = 2, b = 1, c = - 2

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 2 ( - 2 )}{2 \cdot 2}

1^{2} - 4 \cdot 2 (- 2) = 17

1^{2} - 4 \cdot 2 (- 2)

Apply rule

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 2 (- 2)

Apply rule

- (- a) = a

= 1 + 4 \cdot 2 \cdot 2

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= 1 + 16

Add the numbers:

1 + 16 = 17

= 17

u_{1, 2} = \frac{- 1 \pm 17}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- 1 + 17}{2 \cdot 2}, u_{2} = \frac{- 1 - 17}{2 \cdot 2}

u = \frac{- 1 + 17}{2 \cdot 2} : \frac{- 1 + 17}{4}

\frac{- 1 + 17}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 1 + 17}{4}

u = \frac{- 1 - 17}{2 \cdot 2} : \frac{- 1 - 17}{4}

\frac{- 1 - 17}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 1 - 17}{4}

The solutions to the quadratic equation are:

u = \frac{- 1 + 17}{4}, u = \frac{- 1 - 17}{4}

Substitute back

u = sin (x)

sin (x) = \frac{- 1 + 17}{4}, sin (x) = \frac{- 1 - 17}{4}

sin (x) = \frac{- 1 + 17}{4}, sin (x) = \frac{- 1 - 17}{4}

sin (x) = \frac{- 1 + 17}{4} : x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

sin (x) = \frac{- 1 + 17}{4}

Apply trig inverse properties

sin (x) = \frac{- 1 + 17}{4}

General solutions for

sin (x) = \frac{- 1 + 17}{4}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

sin (x) = \frac{- 1 - 17}{4} :

No Solution

sin (x) = \frac{- 1 - 17}{4}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = arcsin (\frac{- 1 + 17}{4}) + 2 πn, x = π - arcsin (\frac{- 1 + 17}{4}) + 2 πn

Show solutions in decimal form

x = 0.89590 \dots + 2 πn, x = π - 0.89590 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for sin(x)-cos(2x)-1=0 ?
The general solution for sin(x)-cos(2x)-1=0 is x=0.89590…+2pin,x=pi-0.89590…+2pin