What is the general solution for 0=1.2209*cos(x)+0.0422*cos(2*x) ?

The general solution for 0=1.2209*cos(x)+0.0422*cos(2*x) is x=1.53630…+2pin,x=2pi-1.53630…+2pin

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0=1.2209cos(x)+0.0422cos(2*x)

Solution

0 = 1.2209 \cdot cos (x) + 0.0422 \cdot cos (2 \cdot x)

Solution

x = 1.53630 \dots + 2 πn, x = 2 π - 1.53630 \dots + 2 πn

Degrees

x = 88.02390 \dots^{\circ} + 36 0^{\circ} n, x = 271.97609 \dots^{\circ} + 36 0^{\circ} n

Solution steps

0 = 1.2209 cos (x) + 0.0422 cos (2 x)

Switch sides

1.2209 cos (x) + 0.0422 cos (2 x) = 0

Rewrite using trig identities

0.0422 cos (2 x) + 1.2209 cos (x)

Use the Double Angle identity:

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

= 0.0422 (2 cos^{2} (x) - 1) + 1.2209 cos (x)

(- 1 + 2 cos^{2} (x)) \cdot 0.0422 + 1.2209 cos (x) = 0

Solve by substitution

(- 1 + 2 cos^{2} (x)) \cdot 0.0422 + 1.2209 cos (x) = 0

Let:

cos (x) = u

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.2209 u = 0

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.2209 u = 0 : u = \frac{- 1.2209 + 1.50484353}{0.1688}, u = \frac{- 1.2209 - 1.50484353}{0.1688}

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.2209 u = 0

Expand

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.2209 u : - 0.0422 + 0.0844 u^{2} + 1.2209 u

(- 1 + 2 u^{2}) \cdot 0.0422 + 1.2209 u

= 0.0422 (- 1 + 2 u^{2}) + 1.2209 u

Expand

0.0422 (- 1 + 2 u^{2}) : - 0.0422 + 0.0844 u^{2}

0.0422 (- 1 + 2 u^{2})

Apply the distributive law:

a (b + c) = ab + a c

a = 0.0422, b = - 1, c = 2 u^{2}

= 0.0422 (- 1) + 0.0422 \cdot 2 u^{2}

Apply minus-plus rules

+ (- a) = - a

= - 1 \cdot 0.0422 + 2 \cdot 0.0422 u^{2}

Simplify

- 1 \cdot 0.0422 + 2 \cdot 0.0422 u^{2} : - 0.0422 + 0.0844 u^{2}

- 1 \cdot 0.0422 + 2 \cdot 0.0422 u^{2}

Multiply the numbers:

1 \cdot 0.0422 = 0.0422

= - 0.0422 + 2 \cdot 0.0422 u^{2}

Multiply the numbers:

2 \cdot 0.0422 = 0.0844

= - 0.0422 + 0.0844 u^{2}

= - 0.0422 + 0.0844 u^{2}

= - 0.0422 + 0.0844 u^{2} + 1.2209 u

- 0.0422 + 0.0844 u^{2} + 1.2209 u = 0

Write in the standard form

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

0.0844 u^{2} + 1.2209 u - 0.0422 = 0

Solve with the quadratic formula

0.0844 u^{2} + 1.2209 u - 0.0422 = 0

Quadratic Equation Formula:

For

a = 0.0844, b = 1.2209, c = - 0.0422

u_{1, 2} = \frac{- 1.2209 \pm 1.220 9 ^{2} - 4 \cdot 0.0844 ( - 0.0422 )}{2 \cdot 0.0844}

u_{1, 2} = \frac{- 1.2209 \pm 1.220 9 ^{2} - 4 \cdot 0.0844 ( - 0.0422 )}{2 \cdot 0.0844}

1.220 9^{2} - 4 \cdot 0.0844 (- 0.0422) = 1.50484353

1.220 9^{2} - 4 \cdot 0.0844 (- 0.0422)

Apply rule

- (- a) = a

= 1.220 9^{2} + 4 \cdot 0.0844 \cdot 0.0422

Multiply the numbers:

4 \cdot 0.0844 \cdot 0.0422 = 0.01424672

= 1.220 9^{2} + 0.01424672

1.220 9^{2} = 1.49059681

= 1.49059681 + 0.01424672

Add the numbers:

1.49059681 + 0.01424672 = 1.50484353

= 1.50484353

u_{1, 2} = \frac{- 1.2209 \pm 1.50484353}{2 \cdot 0.0844}

Separate the solutions

u_{1} = \frac{- 1.2209 + 1.50484353}{2 \cdot 0.0844}, u_{2} = \frac{- 1.2209 - 1.50484353}{2 \cdot 0.0844}

u = \frac{- 1.2209 + 1.50484353}{2 \cdot 0.0844} : \frac{- 1.2209 + 1.50484353}{0.1688}

\frac{- 1.2209 + 1.50484353}{2 \cdot 0.0844}

Multiply the numbers:

2 \cdot 0.0844 = 0.1688

= \frac{- 1.2209 + 1.50484353}{0.1688}

u = \frac{- 1.2209 - 1.50484353}{2 \cdot 0.0844} : \frac{- 1.2209 - 1.50484353}{0.1688}

\frac{- 1.2209 - 1.50484353}{2 \cdot 0.0844}

Multiply the numbers:

2 \cdot 0.0844 = 0.1688

= \frac{- 1.2209 - 1.50484353}{0.1688}

The solutions to the quadratic equation are:

u = \frac{- 1.2209 + 1.50484353}{0.1688}, u = \frac{- 1.2209 - 1.50484353}{0.1688}

Substitute back

u = cos (x)

cos (x) = \frac{- 1.2209 + 1.50484353}{0.1688}, cos (x) = \frac{- 1.2209 - 1.50484353}{0.1688}

cos (x) = \frac{- 1.2209 + 1.50484353}{0.1688}, cos (x) = \frac{- 1.2209 - 1.50484353}{0.1688}

cos (x) = \frac{- 1.2209 + 1.50484353}{0.1688} : x = arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn

cos (x) = \frac{- 1.2209 + 1.50484353}{0.1688}

Apply trig inverse properties

cos (x) = \frac{- 1.2209 + 1.50484353}{0.1688}

General solutions for

cos (x) = \frac{- 1.2209 + 1.50484353}{0.1688}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn

x = arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn

cos (x) = \frac{- 1.2209 - 1.50484353}{0.1688} :

No Solution

cos (x) = \frac{- 1.2209 - 1.50484353}{0.1688}

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn, x = 2 π - arccos (\frac{- 1.2209 + 1.50484353}{0.1688}) + 2 πn

Show solutions in decimal form

x = 1.53630 \dots + 2 πn, x = 2 π - 1.53630 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 0=1.2209*cos(x)+0.0422*cos(2*x) ?
The general solution for 0=1.2209*cos(x)+0.0422*cos(2*x) is x=1.53630…+2pin,x=2pi-1.53630…+2pin