What is the general solution for 8-8sin(x)=5cos^2(x) ?

The general solution for 8-8sin(x)=5cos^2(x) is x= pi/2+2pin,x=0.64350…+2pin,x=pi-0.64350…+2pin

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8-8sin(x)=5cos^2(x)

Solution

8 - 8 sin (x) = 5 cos^{2} (x)

Solution

x = \frac{π}{2} + 2 πn, x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n, x = 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 143.13010 \dots^{\circ} + 36 0^{\circ} n

Solution steps

8 - 8 sin (x) = 5 cos^{2} (x)

Subtract

5 cos^{2} (x)

from both sides

8 - 8 sin (x) - 5 cos^{2} (x) = 0

Rewrite using trig identities

8 - 5 cos^{2} (x) - 8 sin (x)

Use the Pythagorean identity:

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

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

= 8 - 5 (1 - sin^{2} (x)) - 8 sin (x)

Simplify

8 - 5 (1 - sin^{2} (x)) - 8 sin (x) : 5 sin^{2} (x) - 8 sin (x) + 3

8 - 5 (1 - sin^{2} (x)) - 8 sin (x)

Expand

- 5 (1 - sin^{2} (x)) : - 5 + 5 sin^{2} (x)

- 5 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 5, b = 1, c = sin^{2} (x)

= - 5 \cdot 1 - (- 5) sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 5 \cdot 1 + 5 sin^{2} (x)

Multiply the numbers:

5 \cdot 1 = 5

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

= 8 - 5 + 5 sin^{2} (x) - 8 sin (x)

Subtract the numbers:

8 - 5 = 3

= 5 sin^{2} (x) - 8 sin (x) + 3

= 5 sin^{2} (x) - 8 sin (x) + 3

3 + 5 sin^{2} (x) - 8 sin (x) = 0

Solve by substitution

3 + 5 sin^{2} (x) - 8 sin (x) = 0

Let:

sin (x) = u

3 + 5 u^{2} - 8 u = 0

3 + 5 u^{2} - 8 u = 0 : u = 1, u = \frac{3}{5}

3 + 5 u^{2} - 8 u = 0

Write in the standard form

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

5 u^{2} - 8 u + 3 = 0

Solve with the quadratic formula

5 u^{2} - 8 u + 3 = 0

Quadratic Equation Formula:

For

a = 5, b = - 8, c = 3

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

u_{1, 2} = \frac{- ( - 8 ) \pm ( - 8 ) ^{2} - 4 \cdot 5 \cdot 3}{2 \cdot 5}

(- 8)^{2} - 4 \cdot 5 \cdot 3 = 2

(- 8)^{2} - 4 \cdot 5 \cdot 3

Apply exponent rule:

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

n

is even

(- 8)^{2} = 8^{2}

= 8^{2} - 4 \cdot 5 \cdot 3

Multiply the numbers:

4 \cdot 5 \cdot 3 = 60

= 8^{2} - 60

8^{2} = 64

= 64 - 60

Subtract the numbers:

64 - 60 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

u_{1, 2} = \frac{- ( - 8 ) \pm 2}{2 \cdot 5}

Separate the solutions

u_{1} = \frac{- ( - 8 ) + 2}{2 \cdot 5}, u_{2} = \frac{- ( - 8 ) - 2}{2 \cdot 5}

u = \frac{- ( - 8 ) + 2}{2 \cdot 5} : 1

\frac{- ( - 8 ) + 2}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{8 + 2}{2 \cdot 5}

Add the numbers:

8 + 2 = 10

= \frac{10}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- ( - 8 ) - 2}{2 \cdot 5} : \frac{3}{5}

\frac{- ( - 8 ) - 2}{2 \cdot 5}

Apply rule

- (- a) = a

= \frac{8 - 2}{2 \cdot 5}

Subtract the numbers:

8 - 2 = 6

= \frac{6}{2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{6}{10}

Cancel the common factor:

2

= \frac{3}{5}

The solutions to the quadratic equation are:

u = 1, u = \frac{3}{5}

Substitute back

u = sin (x)

sin (x) = 1, sin (x) = \frac{3}{5}

sin (x) = 1, sin (x) = \frac{3}{5}

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

sin (x) = \frac{3}{5} : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

sin (x) = \frac{3}{5}

Apply trig inverse properties

sin (x) = \frac{3}{5}

General solutions for

sin (x) = \frac{3}{5}

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

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Show solutions in decimal form

x = \frac{π}{2} + 2 πn, x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn

Popular Examples

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0 = - \frac{9 π ^{2}}{3200} cos (\frac{π x}{80})

cos^2(a)+2sin(a)+1=0

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tan(x)= 5/15

tan (x) = \frac{5}{15}

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2 sin (x + \frac{π}{9}) = - 1

1+cos(a)=(3+4cos(a))/2

1 + cos (a) = \frac{3 + 4 cos ( a )}{2}

Frequently Asked Questions (FAQ)

What is the general solution for 8-8sin(x)=5cos^2(x) ?
The general solution for 8-8sin(x)=5cos^2(x) is x= pi/2+2pin,x=0.64350…+2pin,x=pi-0.64350…+2pin