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Popular >

8sin^2(x)-3sin(x)-2=0

Solution

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

x = 0.80596 \dots + 2 πn, x = π - 0.80596 \dots + 2 πn, x = - 0.35383 \dots + 2 πn, x = π + 0.35383 \dots + 2 πn

Degrees

x = 46.17848 \dots^{\circ} + 36 0^{\circ} n, x = 133.82151 \dots^{\circ} + 36 0^{\circ} n, x = - 20.27340 \dots^{\circ} + 36 0^{\circ} n, x = 200.27340 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

sin (x) = u

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

8 u^{2} - 3 u - 2 = 0 : u = \frac{3 + 73}{16}, u = \frac{3 - 73}{16}

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = 8, b = - 3, c = - 2

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 8 ( - 2 )}{2 \cdot 8}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 \cdot 8 ( - 2 )}{2 \cdot 8}

(- 3)^{2} - 4 \cdot 8 (- 2) = 73

(- 3)^{2} - 4 \cdot 8 (- 2)

Apply rule

- (- a) = a

= (- 3)^{2} + 4 \cdot 8 \cdot 2

Apply exponent rule:

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

n

is even

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

= 3^{2} + 4 \cdot 8 \cdot 2

Multiply the numbers:

4 \cdot 8 \cdot 2 = 64

= 3^{2} + 64

3^{2} = 9

= 9 + 64

Add the numbers:

9 + 64 = 73

= 73

u_{1, 2} = \frac{- ( - 3 ) \pm 73}{2 \cdot 8}

Separate the solutions

u_{1} = \frac{- ( - 3 ) + 73}{2 \cdot 8}, u_{2} = \frac{- ( - 3 ) - 73}{2 \cdot 8}

u = \frac{- ( - 3 ) + 73}{2 \cdot 8} : \frac{3 + 73}{16}

\frac{- ( - 3 ) + 73}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{3 + 73}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{3 + 73}{16}

u = \frac{- ( - 3 ) - 73}{2 \cdot 8} : \frac{3 - 73}{16}

\frac{- ( - 3 ) - 73}{2 \cdot 8}

Apply rule

- (- a) = a

= \frac{3 - 73}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{3 - 73}{16}

The solutions to the quadratic equation are:

u = \frac{3 + 73}{16}, u = \frac{3 - 73}{16}

Substitute back

u = sin (x)

sin (x) = \frac{3 + 73}{16}, sin (x) = \frac{3 - 73}{16}

sin (x) = \frac{3 + 73}{16}, sin (x) = \frac{3 - 73}{16}

sin (x) = \frac{3 + 73}{16} : x = arcsin (\frac{3 + 73}{16}) + 2 πn, x = π - arcsin (\frac{3 + 73}{16}) + 2 πn

sin (x) = \frac{3 + 73}{16}

Apply trig inverse properties

sin (x) = \frac{3 + 73}{16}

General solutions for

sin (x) = \frac{3 + 73}{16}

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

x = arcsin (\frac{3 + 73}{16}) + 2 πn, x = π - arcsin (\frac{3 + 73}{16}) + 2 πn

x = arcsin (\frac{3 + 73}{16}) + 2 πn, x = π - arcsin (\frac{3 + 73}{16}) + 2 πn

sin (x) = \frac{3 - 73}{16} : x = arcsin (\frac{3 - 73}{16}) + 2 πn, x = π + arcsin (- \frac{3 - 73}{16}) + 2 πn

sin (x) = \frac{3 - 73}{16}

Apply trig inverse properties

sin (x) = \frac{3 - 73}{16}

General solutions for

sin (x) = \frac{3 - 73}{16}

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

x = arcsin (\frac{3 - 73}{16}) + 2 πn, x = π + arcsin (- \frac{3 - 73}{16}) + 2 πn

x = arcsin (\frac{3 - 73}{16}) + 2 πn, x = π + arcsin (- \frac{3 - 73}{16}) + 2 πn

Combine all the solutions

x = arcsin (\frac{3 + 73}{16}) + 2 πn, x = π - arcsin (\frac{3 + 73}{16}) + 2 πn, x = arcsin (\frac{3 - 73}{16}) + 2 πn, x = π + arcsin (- \frac{3 - 73}{16}) + 2 πn

Show solutions in decimal form

x = 0.80596 \dots + 2 πn, x = π - 0.80596 \dots + 2 πn, x = - 0.35383 \dots + 2 πn, x = π + 0.35383 \dots + 2 πn

2 sin^{2} (x) 3 = 7 sin (x), 0 \leq x \leq 2 π

6 sin (x) + 12 = 15

cot^{2} (x) cos (x) = cos (x)

\frac{6}{π} - 4 sin^{2} (x) = 0

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