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Popular >

(218)sin^2(x)+(126)sin(x)-88=0

Solution

(218) sin^{2} (x) + (126) sin (x) - 88 = 0

Solution

x = 0.42135 \dots + 2 πn, x = π - 0.42135 \dots + 2 πn, x = - 1.40923 \dots + 2 πn, x = π + 1.40923 \dots + 2 πn

Degrees

x = 24.14177 \dots^{\circ} + 36 0^{\circ} n, x = 155.85822 \dots^{\circ} + 36 0^{\circ} n, x = - 80.74328 \dots^{\circ} + 36 0^{\circ} n, x = 260.74328 \dots^{\circ} + 36 0^{\circ} n

Solution steps

(218) sin^{2} (x) + (126) sin (x) - 88 = 0

Solve by substitution

218 sin^{2} (x) + 126 sin (x) - 88 = 0

Let:

sin (x) = u

218 u^{2} + 126 u - 88 = 0

218 u^{2} + 126 u - 88 = 0 : u = \frac{- 63 + 13 137}{218}, u = - \frac{63 + 13 137}{218}

218 u^{2} + 126 u - 88 = 0

Solve with the quadratic formula

218 u^{2} + 126 u - 88 = 0

Quadratic Equation Formula:

For

a = 218, b = 126, c = - 88

u_{1, 2} = \frac{- 126 \pm 12 6 ^{2} - 4 \cdot 218 ( - 88 )}{2 \cdot 218}

u_{1, 2} = \frac{- 126 \pm 12 6 ^{2} - 4 \cdot 218 ( - 88 )}{2 \cdot 218}

12 6^{2} - 4 \cdot 218 (- 88) = 26137

12 6^{2} - 4 \cdot 218 (- 88)

Apply rule

- (- a) = a

= 12 6^{2} + 4 \cdot 218 \cdot 88

Multiply the numbers:

4 \cdot 218 \cdot 88 = 76736

= 12 6^{2} + 76736

12 6^{2} = 15876

= 15876 + 76736

Add the numbers:

15876 + 76736 = 92612

= 92612

Prime factorization of

92612 : 2^{2} \cdot 1 3^{2} \cdot 137

92612

92612

divides by

292612 = 46306 \cdot 2

= 2 \cdot 46306

46306

divides by

246306 = 23153 \cdot 2

= 2 \cdot 2 \cdot 23153

23153

divides by

1323153 = 1781 \cdot 13

= 2 \cdot 2 \cdot 13 \cdot 1781

1781

divides by

131781 = 137 \cdot 13

= 2 \cdot 2 \cdot 13 \cdot 13 \cdot 137

2, 13, 137

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 13 \cdot 13 \cdot 137

= 2^{2} \cdot 1 3^{2} \cdot 137

= 2^{2} \cdot 1 3^{2} \cdot 137

Apply radical rule:

= 137 2^{2} 1 3^{2}

Apply radical rule:

2^{2} = 2

= 2137 1 3^{2}

Apply radical rule:

1 3^{2} = 13

= 2 \cdot 13137

Refine

= 26137

u_{1, 2} = \frac{- 126 \pm 26 137}{2 \cdot 218}

Separate the solutions

u_{1} = \frac{- 126 + 26 137}{2 \cdot 218}, u_{2} = \frac{- 126 - 26 137}{2 \cdot 218}

u = \frac{- 126 + 26 137}{2 \cdot 218} : \frac{- 63 + 13 137}{218}

\frac{- 126 + 26 137}{2 \cdot 218}

Multiply the numbers:

2 \cdot 218 = 436

= \frac{- 126 + 26 137}{436}

Factor

- 126 + 26137 : 2 (- 63 + 13137)

- 126 + 26137

Rewrite as

= - 2 \cdot 63 + 2 \cdot 13137

Factor out common term

2

= 2 (- 63 + 13137)

= \frac{2 ( - 63 + 13 137 )}{436}

Cancel the common factor:

2

= \frac{- 63 + 13 137}{218}

u = \frac{- 126 - 26 137}{2 \cdot 218} : - \frac{63 + 13 137}{218}

\frac{- 126 - 26 137}{2 \cdot 218}

Multiply the numbers:

2 \cdot 218 = 436

= \frac{- 126 - 26 137}{436}

Factor

- 126 - 26137 : - 2 (63 + 13137)

- 126 - 26137

Rewrite as

= - 2 \cdot 63 - 2 \cdot 13137

Factor out common term

2

= - 2 (63 + 13137)

= - \frac{2 ( 63 + 13 137 )}{436}

Cancel the common factor:

2

= - \frac{63 + 13 137}{218}

The solutions to the quadratic equation are:

u = \frac{- 63 + 13 137}{218}, u = - \frac{63 + 13 137}{218}

Substitute back

u = sin (x)

sin (x) = \frac{- 63 + 13 137}{218}, sin (x) = - \frac{63 + 13 137}{218}

sin (x) = \frac{- 63 + 13 137}{218}, sin (x) = - \frac{63 + 13 137}{218}

sin (x) = \frac{- 63 + 13 137}{218} : x = arcsin (\frac{- 63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{- 63 + 13 137}{218}) + 2 πn

sin (x) = \frac{- 63 + 13 137}{218}

Apply trig inverse properties

sin (x) = \frac{- 63 + 13 137}{218}

General solutions for

sin (x) = \frac{- 63 + 13 137}{218}

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

x = arcsin (\frac{- 63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{- 63 + 13 137}{218}) + 2 πn

x = arcsin (\frac{- 63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{- 63 + 13 137}{218}) + 2 πn

sin (x) = - \frac{63 + 13 137}{218} : x = arcsin (- \frac{63 + 13 137}{218}) + 2 πn, x = π + arcsin (\frac{63 + 13 137}{218}) + 2 πn

sin (x) = - \frac{63 + 13 137}{218}

Apply trig inverse properties

sin (x) = - \frac{63 + 13 137}{218}

General solutions for

sin (x) = - \frac{63 + 13 137}{218}

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

x = arcsin (- \frac{63 + 13 137}{218}) + 2 πn, x = π + arcsin (\frac{63 + 13 137}{218}) + 2 πn

x = arcsin (- \frac{63 + 13 137}{218}) + 2 πn, x = π + arcsin (\frac{63 + 13 137}{218}) + 2 πn

Combine all the solutions

x = arcsin (\frac{- 63 + 13 137}{218}) + 2 πn, x = π - arcsin (\frac{- 63 + 13 137}{218}) + 2 πn, x = arcsin (- \frac{63 + 13 137}{218}) + 2 πn, x = π + arcsin (\frac{63 + 13 137}{218}) + 2 πn

Show solutions in decimal form

x = 0.42135 \dots + 2 πn, x = π - 0.42135 \dots + 2 πn, x = - 1.40923 \dots + 2 πn, x = π + 1.40923 \dots + 2 πn

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