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Popular >

5sin(x)-3cos(2x)+3=0

Solution

5 sin (x) - 3 cos (2 x) + 3 = 0

Solution

x = 2 πn, x = π + 2 πn, x = - 0.98511 \dots + 2 πn, x = π + 0.98511 \dots + 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n, x = - 56.44269 \dots^{\circ} + 36 0^{\circ} n, x = 236.44269 \dots^{\circ} + 36 0^{\circ} n

Solution steps

5 sin (x) - 3 cos (2 x) + 3 = 0

Rewrite using trig identities

3 - 3 cos (2 x) + 5 sin (x)

Use the Double Angle identity:

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

= 3 - 3 (1 - 2 sin^{2} (x)) + 5 sin (x)

Simplify

3 - 3 (1 - 2 sin^{2} (x)) + 5 sin (x) : 6 sin^{2} (x) + 5 sin (x)

3 - 3 (1 - 2 sin^{2} (x)) + 5 sin (x)

Expand

- 3 (1 - 2 sin^{2} (x)) : - 3 + 6 sin^{2} (x)

- 3 (1 - 2 sin^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = - 3, b = 1, c = 2 sin^{2} (x)

= - 3 \cdot 1 - (- 3) \cdot 2 sin^{2} (x)

Apply minus-plus rules

- (- a) = a

= - 3 \cdot 1 + 3 \cdot 2 sin^{2} (x)

Simplify

- 3 \cdot 1 + 3 \cdot 2 sin^{2} (x) : - 3 + 6 sin^{2} (x)

- 3 \cdot 1 + 3 \cdot 2 sin^{2} (x)

Multiply the numbers:

3 \cdot 1 = 3

= - 3 + 3 \cdot 2 sin^{2} (x)

Multiply the numbers:

3 \cdot 2 = 6

= - 3 + 6 sin^{2} (x)

= - 3 + 6 sin^{2} (x)

= 3 - 3 + 6 sin^{2} (x) + 5 sin (x)

3 - 3 = 0

= 6 sin^{2} (x) + 5 sin (x)

= 6 sin^{2} (x) + 5 sin (x)

5 sin (x) + 6 sin^{2} (x) = 0

Solve by substitution

5 sin (x) + 6 sin^{2} (x) = 0

Let:

sin (x) = u

5 u + 6 u^{2} = 0

5 u + 6 u^{2} = 0 : u = 0, u = - \frac{5}{6}

5 u + 6 u^{2} = 0

Write in the standard form

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

6 u^{2} + 5 u = 0

Solve with the quadratic formula

6 u^{2} + 5 u = 0

Quadratic Equation Formula:

For

a = 6, b = 5, c = 0

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 6 \cdot 0}{2 \cdot 6}

u_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 6 \cdot 0}{2 \cdot 6}

5^{2} - 4 \cdot 6 \cdot 0 = 5

5^{2} - 4 \cdot 6 \cdot 0

Apply rule

0 \cdot a = 0

= 5^{2} - 0

5^{2} - 0 = 5^{2}

= 5^{2}

Apply radical rule:

assuming

a \geq 0

= 5

u_{1, 2} = \frac{- 5 \pm 5}{2 \cdot 6}

Separate the solutions

u_{1} = \frac{- 5 + 5}{2 \cdot 6}, u_{2} = \frac{- 5 - 5}{2 \cdot 6}

u = \frac{- 5 + 5}{2 \cdot 6} : 0

\frac{- 5 + 5}{2 \cdot 6}

Add/Subtract the numbers:

- 5 + 5 = 0

= \frac{0}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{0}{12}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

u = \frac{- 5 - 5}{2 \cdot 6} : - \frac{5}{6}

\frac{- 5 - 5}{2 \cdot 6}

Subtract the numbers:

- 5 - 5 = - 10

= \frac{- 10}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{- 10}{12}

Apply the fraction rule:

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

= - \frac{10}{12}

Cancel the common factor:

2

= - \frac{5}{6}

The solutions to the quadratic equation are:

u = 0, u = - \frac{5}{6}

Substitute back

u = sin (x)

sin (x) = 0, sin (x) = - \frac{5}{6}

sin (x) = 0, sin (x) = - \frac{5}{6}

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

General solutions for

sin (x) = 0

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 = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

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

sin (x) = - \frac{5}{6}

Apply trig inverse properties

sin (x) = - \frac{5}{6}

General solutions for

sin (x) = - \frac{5}{6}

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

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

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

Combine all the solutions

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

Show solutions in decimal form

x = 2 πn, x = π + 2 πn, x = - 0.98511 \dots + 2 πn, x = π + 0.98511 \dots + 2 πn

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