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Popular >

4cos(2x)+10sin(x)-7=0

Solution

4 cos (2 x) + 10 sin (x) - 7 = 0

Solution

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 48.59037 \dots^{\circ} + 36 0^{\circ} n, x = 131.40962 \dots^{\circ} + 36 0^{\circ} n

Solution steps

4 cos (2 x) + 10 sin (x) - 7 = 0

Rewrite using trig identities

- 7 + 10 sin (x) + 4 cos (2 x)

Use the Double Angle identity:

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

= - 7 + 10 sin (x) + 4 (1 - 2 sin^{2} (x))

Simplify

- 7 + 10 sin (x) + 4 (1 - 2 sin^{2} (x)) : 10 sin (x) - 8 sin^{2} (x) - 3

- 7 + 10 sin (x) + 4 (1 - 2 sin^{2} (x))

Expand

4 (1 - 2 sin^{2} (x)) : 4 - 8 sin^{2} (x)

4 (1 - 2 sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = 2 sin^{2} (x)

= 4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

Simplify

4 \cdot 1 - 4 \cdot 2 sin^{2} (x) : 4 - 8 sin^{2} (x)

4 \cdot 1 - 4 \cdot 2 sin^{2} (x)

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 \cdot 2 sin^{2} (x)

Multiply the numbers:

4 \cdot 2 = 8

= 4 - 8 sin^{2} (x)

= 4 - 8 sin^{2} (x)

= - 7 + 10 sin (x) + 4 - 8 sin^{2} (x)

Simplify

- 7 + 10 sin (x) + 4 - 8 sin^{2} (x) : 10 sin (x) - 8 sin^{2} (x) - 3

- 7 + 10 sin (x) + 4 - 8 sin^{2} (x)

Group like terms

= 10 sin (x) - 8 sin^{2} (x) - 7 + 4

Add/Subtract the numbers:

- 7 + 4 = - 3

= 10 sin (x) - 8 sin^{2} (x) - 3

= 10 sin (x) - 8 sin^{2} (x) - 3

= 10 sin (x) - 8 sin^{2} (x) - 3

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

Solve by substitution

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

Let:

sin (x) = u

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

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

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

Write in the standard form

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

- 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 ( - 8 ) ( - 3 )}{2 ( - 8 )}

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

1 0^{2} - 4 (- 8) (- 3) = 2

1 0^{2} - 4 (- 8) (- 3)

Apply rule

- (- a) = a

= 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:

2^{2} = 2

= 2

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

Separate the solutions

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

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

\frac{- 10 + 2}{2 ( - 8 )}

Remove parentheses:

(- a) = - a

= \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 ( - 8 )} : \frac{3}{4}

\frac{- 10 - 2}{2 ( - 8 )}

Remove parentheses:

(- a) = - a

= \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 (x)

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = \frac{1}{2}

General solutions for

sin (x) = \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}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

sin (x) = \frac{3}{4}

Apply trig inverse properties

sin (x) = \frac{3}{4}

General solutions for

sin (x) = \frac{3}{4}

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

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

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

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = arcsin (\frac{3}{4}) + 2 πn, x = π - arcsin (\frac{3}{4}) + 2 πn

Show solutions in decimal form

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = 0.84806 \dots + 2 πn, x = π - 0.84806 \dots + 2 πn

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