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Popular >

-2sin(θ)+3cos(2θ)=0

Solution

- 2 sin (θ) + 3 cos (2 θ) = 0

Solution

θ = - 1.10430 \dots + 2 πn, θ = π + 1.10430 \dots + 2 πn, θ = 0.59416 \dots + 2 πn, θ = π - 0.59416 \dots + 2 πn

Degrees

θ = - 63.27176 \dots^{\circ} + 36 0^{\circ} n, θ = 243.27176 \dots^{\circ} + 36 0^{\circ} n, θ = 34.04310 \dots^{\circ} + 36 0^{\circ} n, θ = 145.95689 \dots^{\circ} + 36 0^{\circ} n

Solution steps

- 2 sin (θ) + 3 cos (2 θ) = 0

Rewrite using trig identities

- 2 sin (θ) + 3 cos (2 θ)

Use the Double Angle identity:

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

= - 2 sin (θ) + 3 (1 - 2 sin^{2} (θ))

(1 - 2 sin^{2} (θ)) \cdot 3 - 2 sin (θ) = 0

Solve by substitution

(1 - 2 sin^{2} (θ)) \cdot 3 - 2 sin (θ) = 0

Let:

sin (θ) = u

(1 - 2 u^{2}) \cdot 3 - 2 u = 0

(1 - 2 u^{2}) \cdot 3 - 2 u = 0 : u = - \frac{1 + 19}{6}, u = \frac{19 - 1}{6}

(1 - 2 u^{2}) \cdot 3 - 2 u = 0

Expand

(1 - 2 u^{2}) \cdot 3 - 2 u : 3 - 6 u^{2} - 2 u

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

= 3 (1 - 2 u^{2}) - 2 u

Expand

3 (1 - 2 u^{2}) : 3 - 6 u^{2}

3 (1 - 2 u^{2})

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = 1, c = 2 u^{2}

= 3 \cdot 1 - 3 \cdot 2 u^{2}

Simplify

3 \cdot 1 - 3 \cdot 2 u^{2} : 3 - 6 u^{2}

3 \cdot 1 - 3 \cdot 2 u^{2}

Multiply the numbers:

3 \cdot 1 = 3

= 3 - 3 \cdot 2 u^{2}

Multiply the numbers:

3 \cdot 2 = 6

= 3 - 6 u^{2}

= 3 - 6 u^{2}

= 3 - 6 u^{2} - 2 u

3 - 6 u^{2} - 2 u = 0

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 6, b = - 2, c = 3

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

u_{1, 2} = \frac{- ( - 2 ) \pm ( - 2 ) ^{2} - 4 ( - 6 ) \cdot 3}{2 ( - 6 )}

(- 2)^{2} - 4 (- 6) \cdot 3 = 219

(- 2)^{2} - 4 (- 6) \cdot 3

Apply rule

- (- a) = a

= (- 2)^{2} + 4 \cdot 6 \cdot 3

Apply exponent rule:

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

n

is even

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

= 2^{2} + 4 \cdot 6 \cdot 3

Multiply the numbers:

4 \cdot 6 \cdot 3 = 72

= 2^{2} + 72

2^{2} = 4

= 4 + 72

Add the numbers:

4 + 72 = 76

= 76

Prime factorization of

76 : 2^{2} \cdot 19

76

76

divides by

276 = 38 \cdot 2

= 2 \cdot 38

38

divides by

238 = 19 \cdot 2

= 2 \cdot 2 \cdot 19

2, 19

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 19

= 2^{2} \cdot 19

= 2^{2} \cdot 19

Apply radical rule:

= 19 2^{2}

Apply radical rule:

2^{2} = 2

= 219

u_{1, 2} = \frac{- ( - 2 ) \pm 2 19}{2 ( - 6 )}

Separate the solutions

u_{1} = \frac{- ( - 2 ) + 2 19}{2 ( - 6 )}, u_{2} = \frac{- ( - 2 ) - 2 19}{2 ( - 6 )}

u = \frac{- ( - 2 ) + 2 19}{2 ( - 6 )} : - \frac{1 + 19}{6}

\frac{- ( - 2 ) + 2 19}{2 ( - 6 )}

Remove parentheses:

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

= \frac{2 + 2 19}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{2 + 2 19}{- 12}

Apply the fraction rule:

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

= - \frac{2 + 2 19}{12}

Cancel

\frac{2 + 2 19}{12} : \frac{1 + 19}{6}

\frac{2 + 2 19}{12}

Factor

2 + 219 : 2 (1 + 19)

2 + 219

Rewrite as

= 2 \cdot 1 + 219

Factor out common term

2

= 2 (1 + 19)

= \frac{2 ( 1 + 19 )}{12}

Cancel the common factor:

2

= \frac{1 + 19}{6}

= - \frac{1 + 19}{6}

u = \frac{- ( - 2 ) - 2 19}{2 ( - 6 )} : \frac{19 - 1}{6}

\frac{- ( - 2 ) - 2 19}{2 ( - 6 )}

Remove parentheses:

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

= \frac{2 - 2 19}{- 2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{2 - 2 19}{- 12}

Apply the fraction rule:

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

2 - 219 = - (219 - 2)

= \frac{2 19 - 2}{12}

Factor

219 - 2 : 2 (19 - 1)

219 - 2

Rewrite as

= 219 - 2 \cdot 1

Factor out common term

2

= 2 (19 - 1)

= \frac{2 ( 19 - 1 )}{12}

Cancel the common factor:

2

= \frac{19 - 1}{6}

The solutions to the quadratic equation are:

u = - \frac{1 + 19}{6}, u = \frac{19 - 1}{6}

Substitute back

u = sin (θ)

sin (θ) = - \frac{1 + 19}{6}, sin (θ) = \frac{19 - 1}{6}

sin (θ) = - \frac{1 + 19}{6}, sin (θ) = \frac{19 - 1}{6}

sin (θ) = - \frac{1 + 19}{6} : θ = arcsin (- \frac{1 + 19}{6}) + 2 πn, θ = π + arcsin (\frac{1 + 19}{6}) + 2 πn

sin (θ) = - \frac{1 + 19}{6}

Apply trig inverse properties

sin (θ) = - \frac{1 + 19}{6}

General solutions for

sin (θ) = - \frac{1 + 19}{6}

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

θ = arcsin (- \frac{1 + 19}{6}) + 2 πn, θ = π + arcsin (\frac{1 + 19}{6}) + 2 πn

θ = arcsin (- \frac{1 + 19}{6}) + 2 πn, θ = π + arcsin (\frac{1 + 19}{6}) + 2 πn

sin (θ) = \frac{19 - 1}{6} : θ = arcsin (\frac{19 - 1}{6}) + 2 πn, θ = π - arcsin (\frac{19 - 1}{6}) + 2 πn

sin (θ) = \frac{19 - 1}{6}

Apply trig inverse properties

sin (θ) = \frac{19 - 1}{6}

General solutions for

sin (θ) = \frac{19 - 1}{6}

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

θ = arcsin (\frac{19 - 1}{6}) + 2 πn, θ = π - arcsin (\frac{19 - 1}{6}) + 2 πn

θ = arcsin (\frac{19 - 1}{6}) + 2 πn, θ = π - arcsin (\frac{19 - 1}{6}) + 2 πn

Combine all the solutions

θ = arcsin (- \frac{1 + 19}{6}) + 2 πn, θ = π + arcsin (\frac{1 + 19}{6}) + 2 πn, θ = arcsin (\frac{19 - 1}{6}) + 2 πn, θ = π - arcsin (\frac{19 - 1}{6}) + 2 πn

Show solutions in decimal form

θ = - 1.10430 \dots + 2 πn, θ = π + 1.10430 \dots + 2 πn, θ = 0.59416 \dots + 2 πn, θ = π - 0.59416 \dots + 2 πn

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