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Popular >

12sin^2(θ)-6sin(θ)=5

Solution

12 sin^{2} (θ) - 6 sin (θ) = 5

Solution

θ = 1.22919 \dots + 2 πn, θ = π - 1.22919 \dots + 2 πn, θ = - 0.45807 \dots + 2 πn, θ = π + 0.45807 \dots + 2 πn

Degrees

θ = 70.42755 \dots^{\circ} + 36 0^{\circ} n, θ = 109.57244 \dots^{\circ} + 36 0^{\circ} n, θ = - 26.24552 \dots^{\circ} + 36 0^{\circ} n, θ = 206.24552 \dots^{\circ} + 36 0^{\circ} n

Solution steps

12 sin^{2} (θ) - 6 sin (θ) = 5

Solve by substitution

12 sin^{2} (θ) - 6 sin (θ) = 5

Let:

sin (θ) = u

12 u^{2} - 6 u = 5

12 u^{2} - 6 u = 5 : u = \frac{3 + 69}{12}, u = \frac{3 - 69}{12}

12 u^{2} - 6 u = 5

Move

5

to the left side

12 u^{2} - 6 u = 5

Subtract

5

from both sides

12 u^{2} - 6 u - 5 = 5 - 5

Simplify

12 u^{2} - 6 u - 5 = 0

12 u^{2} - 6 u - 5 = 0

Solve with the quadratic formula

12 u^{2} - 6 u - 5 = 0

Quadratic Equation Formula:

For

a = 12, b = - 6, c = - 5

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 12 ( - 5 )}{2 \cdot 12}

u_{1, 2} = \frac{- ( - 6 ) \pm ( - 6 ) ^{2} - 4 \cdot 12 ( - 5 )}{2 \cdot 12}

(- 6)^{2} - 4 \cdot 12 (- 5) = 269

(- 6)^{2} - 4 \cdot 12 (- 5)

Apply rule

- (- a) = a

= (- 6)^{2} + 4 \cdot 12 \cdot 5

Apply exponent rule:

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

n

is even

(- 6)^{2} = 6^{2}

= 6^{2} + 4 \cdot 12 \cdot 5

Multiply the numbers:

4 \cdot 12 \cdot 5 = 240

= 6^{2} + 240

6^{2} = 36

= 36 + 240

Add the numbers:

36 + 240 = 276

= 276

Prime factorization of

276 : 2^{2} \cdot 3 \cdot 23

276

276

divides by

2276 = 138 \cdot 2

= 2 \cdot 138

138

divides by

2138 = 69 \cdot 2

= 2 \cdot 2 \cdot 69

69

divides by

369 = 23 \cdot 3

= 2 \cdot 2 \cdot 3 \cdot 23

2, 3, 23

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 3 \cdot 23

= 2^{2} \cdot 3 \cdot 23

= 2^{2} \cdot 3 \cdot 23

Apply radical rule:

= 2^{2} 3 \cdot 23

Apply radical rule:

2^{2} = 2

= 2 3 \cdot 23

Refine

= 269

u_{1, 2} = \frac{- ( - 6 ) \pm 2 69}{2 \cdot 12}

Separate the solutions

u_{1} = \frac{- ( - 6 ) + 2 69}{2 \cdot 12}, u_{2} = \frac{- ( - 6 ) - 2 69}{2 \cdot 12}

u = \frac{- ( - 6 ) + 2 69}{2 \cdot 12} : \frac{3 + 69}{12}

\frac{- ( - 6 ) + 2 69}{2 \cdot 12}

Apply rule

- (- a) = a

= \frac{6 + 2 69}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{6 + 2 69}{24}

Factor

6 + 269 : 2 (3 + 69)

6 + 269

Rewrite as

= 2 \cdot 3 + 269

Factor out common term

2

= 2 (3 + 69)

= \frac{2 ( 3 + 69 )}{24}

Cancel the common factor:

2

= \frac{3 + 69}{12}

u = \frac{- ( - 6 ) - 2 69}{2 \cdot 12} : \frac{3 - 69}{12}

\frac{- ( - 6 ) - 2 69}{2 \cdot 12}

Apply rule

- (- a) = a

= \frac{6 - 2 69}{2 \cdot 12}

Multiply the numbers:

2 \cdot 12 = 24

= \frac{6 - 2 69}{24}

Factor

6 - 269 : 2 (3 - 69)

6 - 269

Rewrite as

= 2 \cdot 3 - 269

Factor out common term

2

= 2 (3 - 69)

= \frac{2 ( 3 - 69 )}{24}

Cancel the common factor:

2

= \frac{3 - 69}{12}

The solutions to the quadratic equation are:

u = \frac{3 + 69}{12}, u = \frac{3 - 69}{12}

Substitute back

u = sin (θ)

sin (θ) = \frac{3 + 69}{12}, sin (θ) = \frac{3 - 69}{12}

sin (θ) = \frac{3 + 69}{12}, sin (θ) = \frac{3 - 69}{12}

sin (θ) = \frac{3 + 69}{12} : θ = arcsin (\frac{3 + 69}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 69}{12}) + 2 πn

sin (θ) = \frac{3 + 69}{12}

Apply trig inverse properties

sin (θ) = \frac{3 + 69}{12}

General solutions for

sin (θ) = \frac{3 + 69}{12}

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

θ = arcsin (\frac{3 + 69}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 69}{12}) + 2 πn

θ = arcsin (\frac{3 + 69}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 69}{12}) + 2 πn

sin (θ) = \frac{3 - 69}{12} : θ = arcsin (\frac{3 - 69}{12}) + 2 πn, θ = π + arcsin (- \frac{3 - 69}{12}) + 2 πn

sin (θ) = \frac{3 - 69}{12}

Apply trig inverse properties

sin (θ) = \frac{3 - 69}{12}

General solutions for

sin (θ) = \frac{3 - 69}{12}

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

θ = arcsin (\frac{3 - 69}{12}) + 2 πn, θ = π + arcsin (- \frac{3 - 69}{12}) + 2 πn

θ = arcsin (\frac{3 - 69}{12}) + 2 πn, θ = π + arcsin (- \frac{3 - 69}{12}) + 2 πn

Combine all the solutions

θ = arcsin (\frac{3 + 69}{12}) + 2 πn, θ = π - arcsin (\frac{3 + 69}{12}) + 2 πn, θ = arcsin (\frac{3 - 69}{12}) + 2 πn, θ = π + arcsin (- \frac{3 - 69}{12}) + 2 πn

Show solutions in decimal form

θ = 1.22919 \dots + 2 πn, θ = π - 1.22919 \dots + 2 πn, θ = - 0.45807 \dots + 2 πn, θ = π + 0.45807 \dots + 2 πn

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