What is the general solution for 2cos^2(x)-7cos(x)=3 ?

The general solution for 2cos^2(x)-7cos(x)=3 is x=1.96708…+2pin,x=-1.96708…+2pin

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Popular >

2cos^2(x)-7cos(x)=3

Solution

2 cos^{2} (x) - 7 cos (x) = 3

Solution

x = 1.96708 \dots + 2 πn, x = - 1.96708 \dots + 2 πn

Degrees

x = 112.70589 \dots^{\circ} + 36 0^{\circ} n, x = - 112.70589 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 cos^{2} (x) - 7 cos (x) = 3

Solve by substitution

2 cos^{2} (x) - 7 cos (x) = 3

Let:

cos (x) = u

2 u^{2} - 7 u = 3

2 u^{2} - 7 u = 3 : u = \frac{7 + 73}{4}, u = \frac{7 - 73}{4}

2 u^{2} - 7 u = 3

Move

3

to the left side

2 u^{2} - 7 u = 3

Subtract

3

from both sides

2 u^{2} - 7 u - 3 = 3 - 3

Simplify

2 u^{2} - 7 u - 3 = 0

2 u^{2} - 7 u - 3 = 0

Solve with the quadratic formula

2 u^{2} - 7 u - 3 = 0

Quadratic Equation Formula:

For

a = 2, b = - 7, c = - 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 ( - 3 )}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 (- 3) = 73

(- 7)^{2} - 4 \cdot 2 (- 3)

Apply rule

- (- a) = a

= (- 7)^{2} + 4 \cdot 2 \cdot 3

Apply exponent rule:

(- a)^{n} = a^{n},

n

is even

(- 7)^{2} = 7^{2}

= 7^{2} + 4 \cdot 2 \cdot 3

Multiply the numbers:

4 \cdot 2 \cdot 3 = 24

= 7^{2} + 24

7^{2} = 49

= 49 + 24

Add the numbers:

49 + 24 = 73

= 73

u_{1, 2} = \frac{- ( - 7 ) \pm 73}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 7 ) + 73}{2 \cdot 2}, u_{2} = \frac{- ( - 7 ) - 73}{2 \cdot 2}

u = \frac{- ( - 7 ) + 73}{2 \cdot 2} : \frac{7 + 73}{4}

\frac{- ( - 7 ) + 73}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{7 + 73}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{7 + 73}{4}

u = \frac{- ( - 7 ) - 73}{2 \cdot 2} : \frac{7 - 73}{4}

\frac{- ( - 7 ) - 73}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{7 - 73}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{7 - 73}{4}

The solutions to the quadratic equation are:

u = \frac{7 + 73}{4}, u = \frac{7 - 73}{4}

Substitute back

u = cos (x)

cos (x) = \frac{7 + 73}{4}, cos (x) = \frac{7 - 73}{4}

cos (x) = \frac{7 + 73}{4}, cos (x) = \frac{7 - 73}{4}

cos (x) = \frac{7 + 73}{4} :

No Solution

cos (x) = \frac{7 + 73}{4}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{7 - 73}{4} : x = arccos (\frac{7 - 73}{4}) + 2 πn, x = - arccos (\frac{7 - 73}{4}) + 2 πn

cos (x) = \frac{7 - 73}{4}

Apply trig inverse properties

cos (x) = \frac{7 - 73}{4}

General solutions for

cos (x) = \frac{7 - 73}{4}

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

x = arccos (\frac{7 - 73}{4}) + 2 πn, x = - arccos (\frac{7 - 73}{4}) + 2 πn

x = arccos (\frac{7 - 73}{4}) + 2 πn, x = - arccos (\frac{7 - 73}{4}) + 2 πn

Combine all the solutions

x = arccos (\frac{7 - 73}{4}) + 2 πn, x = - arccos (\frac{7 - 73}{4}) + 2 πn

Show solutions in decimal form

x = 1.96708 \dots + 2 πn, x = - 1.96708 \dots + 2 πn

Popular Examples

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tan (x) = tan (2 x)

2cos(x)= 1/2

2 cos (x) = \frac{1}{2}

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

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

cos(x)+0.65=0

cos (x) + 0.65 = 0

sin^2(x)=0.12

sin^{2} (x) = 0.12

Frequently Asked Questions (FAQ)

What is the general solution for 2cos^2(x)-7cos(x)=3 ?
The general solution for 2cos^2(x)-7cos(x)=3 is x=1.96708…+2pin,x=-1.96708…+2pin