What is the general solution for -5cos(x)=2sin^2(x)+4 ?

The general solution for -5cos(x)=2sin^2(x)+4 is x=2.65944…+2pin,x=-2.65944…+2pin

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Popular >

-5cos(x)=2sin^2(x)+4

Solution

- 5 cos (x) = 2 sin^{2} (x) + 4

Solution

x = 2.65944 \dots + 2 πn, x = - 2.65944 \dots + 2 πn

Degrees

x = 152.37494 \dots^{\circ} + 36 0^{\circ} n, x = - 152.37494 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 5 cos (x) = 2 sin^{2} (x) + 4

Subtract

2 sin^{2} (x) + 4

from both sides

- 5 cos (x) - 2 sin^{2} (x) - 4 = 0

Rewrite using trig identities

- 4 - 2 sin^{2} (x) - 5 cos (x)

Use the Pythagorean identity:

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

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

= - 4 - 2 (1 - cos^{2} (x)) - 5 cos (x)

Simplify

- 4 - 2 (1 - cos^{2} (x)) - 5 cos (x) : 2 cos^{2} (x) - 5 cos (x) - 6

- 4 - 2 (1 - cos^{2} (x)) - 5 cos (x)

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = - 2, b = 1, c = cos^{2} (x)

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

Apply minus-plus rules

- (- a) = a

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

Multiply the numbers:

2 \cdot 1 = 2

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

= - 4 - 2 + 2 cos^{2} (x) - 5 cos (x)

Subtract the numbers:

- 4 - 2 = - 6

= 2 cos^{2} (x) - 5 cos (x) - 6

= 2 cos^{2} (x) - 5 cos (x) - 6

- 6 + 2 cos^{2} (x) - 5 cos (x) = 0

Solve by substitution

- 6 + 2 cos^{2} (x) - 5 cos (x) = 0

Let:

cos (x) = u

- 6 + 2 u^{2} - 5 u = 0

- 6 + 2 u^{2} - 5 u = 0 : u = \frac{5 + 73}{4}, u = \frac{5 - 73}{4}

- 6 + 2 u^{2} - 5 u = 0

Write in the standard form

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

2 u^{2} - 5 u - 6 = 0

Solve with the quadratic formula

2 u^{2} - 5 u - 6 = 0

Quadratic Equation Formula:

For

a = 2, b = - 5, c = - 6

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 ( - 6 )}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 2 ( - 6 )}{2 \cdot 2}

(- 5)^{2} - 4 \cdot 2 (- 6) = 73

(- 5)^{2} - 4 \cdot 2 (- 6)

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 2 \cdot 6

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 2 \cdot 6

Multiply the numbers:

4 \cdot 2 \cdot 6 = 48

= 5^{2} + 48

5^{2} = 25

= 25 + 48

Add the numbers:

25 + 48 = 73

= 73

u_{1, 2} = \frac{- ( - 5 ) \pm 73}{2 \cdot 2}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 73}{2 \cdot 2}, u_{2} = \frac{- ( - 5 ) - 73}{2 \cdot 2}

u = \frac{- ( - 5 ) + 73}{2 \cdot 2} : \frac{5 + 73}{4}

\frac{- ( - 5 ) + 73}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 + 73}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{5 + 73}{4}

u = \frac{- ( - 5 ) - 73}{2 \cdot 2} : \frac{5 - 73}{4}

\frac{- ( - 5 ) - 73}{2 \cdot 2}

Apply rule

- (- a) = a

= \frac{5 - 73}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{5 - 73}{4}

The solutions to the quadratic equation are:

u = \frac{5 + 73}{4}, u = \frac{5 - 73}{4}

Substitute back

u = cos (x)

cos (x) = \frac{5 + 73}{4}, cos (x) = \frac{5 - 73}{4}

cos (x) = \frac{5 + 73}{4}, cos (x) = \frac{5 - 73}{4}

cos (x) = \frac{5 + 73}{4} :

No Solution

cos (x) = \frac{5 + 73}{4}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{5 - 73}{4} : x = arccos (\frac{5 - 73}{4}) + 2 πn, x = - arccos (\frac{5 - 73}{4}) + 2 πn

cos (x) = \frac{5 - 73}{4}

Apply trig inverse properties

cos (x) = \frac{5 - 73}{4}

General solutions for

cos (x) = \frac{5 - 73}{4}

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

x = arccos (\frac{5 - 73}{4}) + 2 πn, x = - arccos (\frac{5 - 73}{4}) + 2 πn

x = arccos (\frac{5 - 73}{4}) + 2 πn, x = - arccos (\frac{5 - 73}{4}) + 2 πn

Combine all the solutions

x = arccos (\frac{5 - 73}{4}) + 2 πn, x = - arccos (\frac{5 - 73}{4}) + 2 πn

Show solutions in decimal form

x = 2.65944 \dots + 2 πn, x = - 2.65944 \dots + 2 πn

Popular Examples

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5 sin (θ + 0.4636) = 1

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solvefor y,arctan(y)=(t^2)/2 solve for

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cos(-x)=0

cos (- x) = 0

2tan^2(x)+1=0

2 tan^{2} (x) + 1 = 0

Frequently Asked Questions (FAQ)

What is the general solution for -5cos(x)=2sin^2(x)+4 ?
The general solution for -5cos(x)=2sin^2(x)+4 is x=2.65944…+2pin,x=-2.65944…+2pin