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

The general solution for 2cos^2(θ)-3sin(θ)+2=0 is θ=1.01746…+2pin,θ=pi-1.01746…+2pin

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Popular >

2cos^2(θ)-3sin(θ)+2=0

Solution

2 cos^{2} (θ) - 3 sin (θ) + 2 = 0

Solution

θ = 1.01746 \dots + 2 πn, θ = π - 1.01746 \dots + 2 πn

Degrees

θ = 58.29672 \dots^{\circ} + 36 0^{\circ} n, θ = 121.70327 \dots^{\circ} + 36 0^{\circ} n

Solution steps

2 cos^{2} (θ) - 3 sin (θ) + 2 = 0

Rewrite using trig identities

2 + 2 cos^{2} (θ) - 3 sin (θ)

Use the Pythagorean identity:

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

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

= 2 + 2 (1 - sin^{2} (θ)) - 3 sin (θ)

Simplify

2 + 2 (1 - sin^{2} (θ)) - 3 sin (θ) : - 2 sin^{2} (θ) - 3 sin (θ) + 4

2 + 2 (1 - sin^{2} (θ)) - 3 sin (θ)

Expand

2 (1 - sin^{2} (θ)) : 2 - 2 sin^{2} (θ)

2 (1 - sin^{2} (θ))

Apply the distributive law:

a (b - c) = ab - a c

a = 2, b = 1, c = sin^{2} (θ)

= 2 \cdot 1 - 2 sin^{2} (θ)

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 sin^{2} (θ)

= 2 + 2 - 2 sin^{2} (θ) - 3 sin (θ)

Add the numbers:

2 + 2 = 4

= - 2 sin^{2} (θ) - 3 sin (θ) + 4

= - 2 sin^{2} (θ) - 3 sin (θ) + 4

4 - 2 sin^{2} (θ) - 3 sin (θ) = 0

Solve by substitution

4 - 2 sin^{2} (θ) - 3 sin (θ) = 0

Let:

sin (θ) = u

4 - 2 u^{2} - 3 u = 0

4 - 2 u^{2} - 3 u = 0 : u = - \frac{3 + 41}{4}, u = \frac{41 - 3}{4}

4 - 2 u^{2} - 3 u = 0

Write in the standard form

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

- 2 u^{2} - 3 u + 4 = 0

Solve with the quadratic formula

- 2 u^{2} - 3 u + 4 = 0

Quadratic Equation Formula:

For

a = - 2, b = - 3, c = 4

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

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

(- 3)^{2} - 4 (- 2) \cdot 4 = 41

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 2 \cdot 4 = 32

= 3^{2} + 32

3^{2} = 9

= 9 + 32

Add the numbers:

9 + 32 = 41

= 41

u_{1, 2} = \frac{- ( - 3 ) \pm 41}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 41}{2 ( - 2 )}, u_{2} = \frac{- ( - 3 ) - 41}{2 ( - 2 )}

u = \frac{- ( - 3 ) + 41}{2 ( - 2 )} : - \frac{3 + 41}{4}

\frac{- ( - 3 ) + 41}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 + 41}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 + 41}{- 4}

Apply the fraction rule:

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

= - \frac{3 + 41}{4}

u = \frac{- ( - 3 ) - 41}{2 ( - 2 )} : \frac{41 - 3}{4}

\frac{- ( - 3 ) - 41}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 - 41}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3 - 41}{- 4}

Apply the fraction rule:

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

3 - 41 = - (41 - 3)

= \frac{41 - 3}{4}

The solutions to the quadratic equation are:

u = - \frac{3 + 41}{4}, u = \frac{41 - 3}{4}

Substitute back

u = sin (θ)

sin (θ) = - \frac{3 + 41}{4}, sin (θ) = \frac{41 - 3}{4}

sin (θ) = - \frac{3 + 41}{4}, sin (θ) = \frac{41 - 3}{4}

sin (θ) = - \frac{3 + 41}{4} :

No Solution

sin (θ) = - \frac{3 + 41}{4}

- 1 \leq sin (x) \leq 1

No Solution

sin (θ) = \frac{41 - 3}{4} : θ = arcsin (\frac{41 - 3}{4}) + 2 πn, θ = π - arcsin (\frac{41 - 3}{4}) + 2 πn

sin (θ) = \frac{41 - 3}{4}

Apply trig inverse properties

sin (θ) = \frac{41 - 3}{4}

General solutions for

sin (θ) = \frac{41 - 3}{4}

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

θ = arcsin (\frac{41 - 3}{4}) + 2 πn, θ = π - arcsin (\frac{41 - 3}{4}) + 2 πn

θ = arcsin (\frac{41 - 3}{4}) + 2 πn, θ = π - arcsin (\frac{41 - 3}{4}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{41 - 3}{4}) + 2 πn, θ = π - arcsin (\frac{41 - 3}{4}) + 2 πn

Show solutions in decimal form

θ = 1.01746 \dots + 2 πn, θ = π - 1.01746 \dots + 2 πn

Popular Examples

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36 cos^{2} (x) - 9 = 0

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

Frequently Asked Questions (FAQ)

What is the general solution for 2cos^2(θ)-3sin(θ)+2=0 ?
The general solution for 2cos^2(θ)-3sin(θ)+2=0 is θ=1.01746…+2pin,θ=pi-1.01746…+2pin