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

The general solution for sin^2(x)=3cos(x)-2 is x=0.65788…+2pin,x=2pi-0.65788…+2pin

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

Solution

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

Solution

x = 0.65788 \dots + 2 πn, x = 2 π - 0.65788 \dots + 2 πn

Degrees

x = 37.69397 \dots^{\circ} + 36 0^{\circ} n, x = 322.30602 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

3 cos (x) - 2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Solve by substitution

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

Let:

cos (x) = u

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

3 - u^{2} - 3 u = 0 : u = - \frac{3 + 21}{2}, u = \frac{21 - 3}{2}

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

(- 3)^{2} - 4 (- 1) \cdot 3 = 21

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

= 3^{2} + 4 \cdot 1 \cdot 3

Multiply the numbers:

4 \cdot 1 \cdot 3 = 12

= 3^{2} + 12

3^{2} = 9

= 9 + 12

Add the numbers:

9 + 12 = 21

= 21

u_{1, 2} = \frac{- ( - 3 ) \pm 21}{2 ( - 1 )}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 21}{2 ( - 1 )}, u_{2} = \frac{- ( - 3 ) - 21}{2 ( - 1 )}

u = \frac{- ( - 3 ) + 21}{2 ( - 1 )} : - \frac{3 + 21}{2}

\frac{- ( - 3 ) + 21}{2 ( - 1 )}

Remove parentheses:

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

= \frac{3 + 21}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 + 21}{- 2}

Apply the fraction rule:

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

= - \frac{3 + 21}{2}

u = \frac{- ( - 3 ) - 21}{2 ( - 1 )} : \frac{21 - 3}{2}

\frac{- ( - 3 ) - 21}{2 ( - 1 )}

Remove parentheses:

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

= \frac{3 - 21}{- 2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{3 - 21}{- 2}

Apply the fraction rule:

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

3 - 21 = - (21 - 3)

= \frac{21 - 3}{2}

The solutions to the quadratic equation are:

u = - \frac{3 + 21}{2}, u = \frac{21 - 3}{2}

Substitute back

u = cos (x)

cos (x) = - \frac{3 + 21}{2}, cos (x) = \frac{21 - 3}{2}

cos (x) = - \frac{3 + 21}{2}, cos (x) = \frac{21 - 3}{2}

cos (x) = - \frac{3 + 21}{2} :

No Solution

cos (x) = - \frac{3 + 21}{2}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{21 - 3}{2} : x = arccos (\frac{21 - 3}{2}) + 2 πn, x = 2 π - arccos (\frac{21 - 3}{2}) + 2 πn

cos (x) = \frac{21 - 3}{2}

Apply trig inverse properties

cos (x) = \frac{21 - 3}{2}

General solutions for

cos (x) = \frac{21 - 3}{2}

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

x = arccos (\frac{21 - 3}{2}) + 2 πn, x = 2 π - arccos (\frac{21 - 3}{2}) + 2 πn

x = arccos (\frac{21 - 3}{2}) + 2 πn, x = 2 π - arccos (\frac{21 - 3}{2}) + 2 πn

Combine all the solutions

x = arccos (\frac{21 - 3}{2}) + 2 πn, x = 2 π - arccos (\frac{21 - 3}{2}) + 2 πn

Show solutions in decimal form

x = 0.65788 \dots + 2 πn, x = 2 π - 0.65788 \dots + 2 πn

Popular Examples

4sin(2x+pi/6)=2

4 sin (2 x + \frac{π}{6}) = 2

8cos(θ)=3-4cos(θ)

8 cos (θ) = 3 - 4 cos (θ)

cos(θ)=(sqrt(3))/2 ,sin(θ)

cos (θ) = \frac{3}{2}, sin (θ)

sin(θ)=0.788

sin (θ) = 0.788

arctan(θ)=-sqrt(3)

arctan (θ) = - 3

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)=3cos(x)-2 ?
The general solution for sin^2(x)=3cos(x)-2 is x=0.65788…+2pin,x=2pi-0.65788…+2pin