What is the general solution for 3cos^2(x)-sin(x)-1=0 ?

The general solution for 3cos^2(x)-sin(x)-1=0 is x=(3pi)/2+2pin,x=0.72972…+2pin,x=pi-0.72972…+2pin

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3cos^2(x)-sin(x)-1=0

Solution

3 cos^{2} (x) - sin (x) - 1 = 0

Solution

x = \frac{3 π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n, x = 41.81031 \dots^{\circ} + 36 0^{\circ} n, x = 138.18968 \dots^{\circ} + 36 0^{\circ} n

Solution steps

3 cos^{2} (x) - sin (x) - 1 = 0

Rewrite using trig identities

- 1 - sin (x) + 3 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 1 - sin (x) + 3 (1 - sin^{2} (x))

Simplify

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

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

Expand

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

3 (1 - sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = sin^{2} (x)

= 3 \cdot 1 - 3 sin^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 sin^{2} (x)

= - 1 - sin (x) + 3 - 3 sin^{2} (x)

Simplify

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

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

Group like terms

= - sin (x) - 3 sin^{2} (x) - 1 + 3

Add/Subtract the numbers:

- 1 + 3 = 2

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 3, b = - 1, c = 2

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

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

(- 1)^{2} - 4 (- 3) \cdot 2 = 5

(- 1)^{2} - 4 (- 3) \cdot 2

Apply rule

- (- a) = a

= (- 1)^{2} + 4 \cdot 3 \cdot 2

(- 1)^{2} = 1

(- 1)^{2}

Apply exponent rule:

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

n

is even

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

= 1^{2}

Apply rule

1^{a} = 1

= 1

4 \cdot 3 \cdot 2 = 24

4 \cdot 3 \cdot 2

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 24

= 1 + 24

Add the numbers:

1 + 24 = 25

= 25

Factor the number:

25 = 5^{2}

= 5^{2}

Apply radical rule:

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 1 ) \pm 5}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- ( - 1 ) + 5}{2 ( - 3 )}, u_{2} = \frac{- ( - 1 ) - 5}{2 ( - 3 )}

u = \frac{- ( - 1 ) + 5}{2 ( - 3 )} : - 1

\frac{- ( - 1 ) + 5}{2 ( - 3 )}

Remove parentheses:

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

= \frac{1 + 5}{- 2 \cdot 3}

Add the numbers:

1 + 5 = 6

= \frac{6}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{- 6}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{6}{6}

Apply rule

\frac{a}{a} = 1

= - 1

u = \frac{- ( - 1 ) - 5}{2 ( - 3 )} : \frac{2}{3}

\frac{- ( - 1 ) - 5}{2 ( - 3 )}

Remove parentheses:

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

= \frac{1 - 5}{- 2 \cdot 3}

Subtract the numbers:

1 - 5 = - 4

= \frac{- 4}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 4}{- 6}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{6}

Cancel the common factor:

2

= \frac{2}{3}

The solutions to the quadratic equation are:

u = - 1, u = \frac{2}{3}

Substitute back

u = sin (x)

sin (x) = - 1, sin (x) = \frac{2}{3}

sin (x) = - 1, sin (x) = \frac{2}{3}

sin (x) = - 1 : x = \frac{3 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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

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

sin (x) = \frac{2}{3}

Apply trig inverse properties

sin (x) = \frac{2}{3}

General solutions for

sin (x) = \frac{2}{3}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = \frac{3 π}{2} + 2 πn, x = 0.72972 \dots + 2 πn, x = π - 0.72972 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos^2(x)-sin(x)-1=0 ?
The general solution for 3cos^2(x)-sin(x)-1=0 is x=(3pi)/2+2pin,x=0.72972…+2pin,x=pi-0.72972…+2pin