What is the general solution for 3sin(x)sin(x)=5cos(x)-2 ?

The general solution for 3sin(x)sin(x)=5cos(x)-2 is x=0.79082…+2pin,x=2pi-0.79082…+2pin

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

Solution

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

Solution

x = 0.79082 \dots + 2 πn, x = 2 π - 0.79082 \dots + 2 πn

Degrees

x = 45.31106 \dots^{\circ} + 36 0^{\circ} n, x = 314.68893 \dots^{\circ} + 36 0^{\circ} n

Solution steps

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

Subtract

5 cos (x) - 2

from both sides

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

3 (1 - cos^{2} (x)) : 3 - 3 cos^{2} (x)

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

3 \cdot 1 = 3

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

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

Add the numbers:

2 + 3 = 5

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

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

5 - 3 cos^{2} (x) - 5 cos (x) = 0

Solve by substitution

5 - 3 cos^{2} (x) - 5 cos (x) = 0

Let:

cos (x) = u

5 - 3 u^{2} - 5 u = 0

5 - 3 u^{2} - 5 u = 0 : u = - \frac{5 + 85}{6}, u = \frac{85 - 5}{6}

5 - 3 u^{2} - 5 u = 0

Write in the standard form

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

- 3 u^{2} - 5 u + 5 = 0

Solve with the quadratic formula

- 3 u^{2} - 5 u + 5 = 0

Quadratic Equation Formula:

For

a = - 3, b = - 5, c = 5

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 3 ) \cdot 5}{2 ( - 3 )}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 ( - 3 ) \cdot 5}{2 ( - 3 )}

(- 5)^{2} - 4 (- 3) \cdot 5 = 85

(- 5)^{2} - 4 (- 3) \cdot 5

Apply rule

- (- a) = a

= (- 5)^{2} + 4 \cdot 3 \cdot 5

Apply exponent rule:

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

n

is even

(- 5)^{2} = 5^{2}

= 5^{2} + 4 \cdot 3 \cdot 5

Multiply the numbers:

4 \cdot 3 \cdot 5 = 60

= 5^{2} + 60

5^{2} = 25

= 25 + 60

Add the numbers:

25 + 60 = 85

= 85

u_{1, 2} = \frac{- ( - 5 ) \pm 85}{2 ( - 3 )}

Separate the solutions

u_{1} = \frac{- ( - 5 ) + 85}{2 ( - 3 )}, u_{2} = \frac{- ( - 5 ) - 85}{2 ( - 3 )}

u = \frac{- ( - 5 ) + 85}{2 ( - 3 )} : - \frac{5 + 85}{6}

\frac{- ( - 5 ) + 85}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{5 + 85}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5 + 85}{- 6}

Apply the fraction rule:

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

= - \frac{5 + 85}{6}

u = \frac{- ( - 5 ) - 85}{2 ( - 3 )} : \frac{85 - 5}{6}

\frac{- ( - 5 ) - 85}{2 ( - 3 )}

Remove parentheses:

(- a) = - a, - (- a) = a

= \frac{5 - 85}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{5 - 85}{- 6}

Apply the fraction rule:

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

5 - 85 = - (85 - 5)

= \frac{85 - 5}{6}

The solutions to the quadratic equation are:

u = - \frac{5 + 85}{6}, u = \frac{85 - 5}{6}

Substitute back

u = cos (x)

cos (x) = - \frac{5 + 85}{6}, cos (x) = \frac{85 - 5}{6}

cos (x) = - \frac{5 + 85}{6}, cos (x) = \frac{85 - 5}{6}

cos (x) = - \frac{5 + 85}{6} :

No Solution

cos (x) = - \frac{5 + 85}{6}

- 1 \leq cos (x) \leq 1

No Solution

cos (x) = \frac{85 - 5}{6} : x = arccos (\frac{85 - 5}{6}) + 2 πn, x = 2 π - arccos (\frac{85 - 5}{6}) + 2 πn

cos (x) = \frac{85 - 5}{6}

Apply trig inverse properties

cos (x) = \frac{85 - 5}{6}

General solutions for

cos (x) = \frac{85 - 5}{6}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{85 - 5}{6}) + 2 πn, x = 2 π - arccos (\frac{85 - 5}{6}) + 2 πn

x = arccos (\frac{85 - 5}{6}) + 2 πn, x = 2 π - arccos (\frac{85 - 5}{6}) + 2 πn

Combine all the solutions

x = arccos (\frac{85 - 5}{6}) + 2 πn, x = 2 π - arccos (\frac{85 - 5}{6}) + 2 πn

Show solutions in decimal form

x = 0.79082 \dots + 2 πn, x = 2 π - 0.79082 \dots + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 3sin(x)sin(x)=5cos(x)-2 ?
The general solution for 3sin(x)sin(x)=5cos(x)-2 is x=0.79082…+2pin,x=2pi-0.79082…+2pin