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Popular Trigonometry >

8sin^2(w)+10sin(w)+3=0

Solution

8 sin^{2} (w) + 10 sin (w) + 3 = 0

Solution

w = \frac{7 π}{6} + 2 πn, w = \frac{11 π}{6} + 2 πn, w = - 0.84806 \dots + 2 πn, w = π + 0.84806 \dots + 2 πn

Degrees

w = 21 0^{\circ} + 36 0^{\circ} n, w = 33 0^{\circ} + 36 0^{\circ} n, w = - 48.59037 \dots^{\circ} + 36 0^{\circ} n, w = 228.59037 \dots^{\circ} + 36 0^{\circ} n

Solution steps

8 sin^{2} (w) + 10 sin (w) + 3 = 0

Solve by substitution

8 sin^{2} (w) + 10 sin (w) + 3 = 0

Let:

sin (w) = u

8 u^{2} + 10 u + 3 = 0

8 u^{2} + 10 u + 3 = 0 : u = - \frac{1}{2}, u = - \frac{3}{4}

8 u^{2} + 10 u + 3 = 0

Solve with the quadratic formula

8 u^{2} + 10 u + 3 = 0

Quadratic Equation Formula:

For

a = 8, b = 10, c = 3

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 8 \cdot 3}{2 \cdot 8}

u_{1, 2} = \frac{- 10 \pm 1 0 ^{2} - 4 \cdot 8 \cdot 3}{2 \cdot 8}

1 0^{2} - 4 \cdot 8 \cdot 3 = 2

1 0^{2} - 4 \cdot 8 \cdot 3

Multiply the numbers:

4 \cdot 8 \cdot 3 = 96

= 1 0^{2} - 96

1 0^{2} = 100

= 100 - 96

Subtract the numbers:

100 - 96 = 4

= 4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

n a^{n} = a

2^{2} = 2

= 2

u_{1, 2} = \frac{- 10 \pm 2}{2 \cdot 8}

Separate the solutions

u_{1} = \frac{- 10 + 2}{2 \cdot 8}, u_{2} = \frac{- 10 - 2}{2 \cdot 8}

u = \frac{- 10 + 2}{2 \cdot 8} : - \frac{1}{2}

\frac{- 10 + 2}{2 \cdot 8}

Add/Subtract the numbers:

- 10 + 2 = - 8

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

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 8}{16}

Apply the fraction rule:

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

= - \frac{8}{16}

Cancel the common factor:

8

= - \frac{1}{2}

u = \frac{- 10 - 2}{2 \cdot 8} : - \frac{3}{4}

\frac{- 10 - 2}{2 \cdot 8}

Subtract the numbers:

- 10 - 2 = - 12

= \frac{- 12}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{- 12}{16}

Apply the fraction rule:

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

= - \frac{12}{16}

Cancel the common factor:

4

= - \frac{3}{4}

The solutions to the quadratic equation are:

u = - \frac{1}{2}, u = - \frac{3}{4}

Substitute back

u = sin (w)

sin (w) = - \frac{1}{2}, sin (w) = - \frac{3}{4}

sin (w) = - \frac{1}{2}, sin (w) = - \frac{3}{4}

sin (w) = - \frac{1}{2} : w = \frac{7 π}{6} + 2 πn, w = \frac{11 π}{6} + 2 πn

sin (w) = - \frac{1}{2}

General solutions for

sin (w) = - \frac{1}{2}

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

w = \frac{7 π}{6} + 2 πn, w = \frac{11 π}{6} + 2 πn

w = \frac{7 π}{6} + 2 πn, w = \frac{11 π}{6} + 2 πn

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

sin (w) = - \frac{3}{4}

Apply trig inverse properties

sin (w) = - \frac{3}{4}

General solutions for

sin (w) = - \frac{3}{4}

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

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

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

Combine all the solutions

w = \frac{7 π}{6} + 2 πn, w = \frac{11 π}{6} + 2 πn, w = arcsin (- \frac{3}{4}) + 2 πn, w = π + arcsin (\frac{3}{4}) + 2 πn

Show solutions in decimal form

w = \frac{7 π}{6} + 2 πn, w = \frac{11 π}{6} + 2 πn, w = - 0.84806 \dots + 2 πn, w = π + 0.84806 \dots + 2 πn

Graph

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