What is the general solution for 5cos^2(x)+3sin(x)-3=0 ?

The general solution for 5cos^2(x)+3sin(x)-3=0 is x=-0.41151…+2pin,x=pi+0.41151…+2pin,x= pi/2+2pin

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Popular >

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

Solution

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

Solution

x = - 0.41151 \dots + 2 πn, x = π + 0.41151 \dots + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = - 23.57817 \dots^{\circ} + 36 0^{\circ} n, x = 203.57817 \dots^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

a = 5, b = 1, c = sin^{2} (x)

= 5 \cdot 1 - 5 sin^{2} (x)

Multiply the numbers:

5 \cdot 1 = 5

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

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

Simplify

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

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

Group like terms

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

Add/Subtract the numbers:

- 3 + 5 = 2

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

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

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

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

Solve by substitution

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

Let:

sin (x) = u

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

2 + 3 u - 5 u^{2} = 0 : u = - \frac{2}{5}, u = 1

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

a = - 5, b = 3, c = 2

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

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

3^{2} - 4 (- 5) \cdot 2 = 7

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 5 \cdot 2 = 40

= 3^{2} + 40

3^{2} = 9

= 9 + 40

Add the numbers:

9 + 40 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

7^{2} = 7

= 7

u_{1, 2} = \frac{- 3 \pm 7}{2 ( - 5 )}

Separate the solutions

u_{1} = \frac{- 3 + 7}{2 ( - 5 )}, u_{2} = \frac{- 3 - 7}{2 ( - 5 )}

u = \frac{- 3 + 7}{2 ( - 5 )} : - \frac{2}{5}

\frac{- 3 + 7}{2 ( - 5 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 + 7}{- 2 \cdot 5}

Add/Subtract the numbers:

- 3 + 7 = 4

= \frac{4}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{4}{- 10}

Apply the fraction rule:

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

= - \frac{4}{10}

Cancel the common factor:

2

= - \frac{2}{5}

u = \frac{- 3 - 7}{2 ( - 5 )} : 1

\frac{- 3 - 7}{2 ( - 5 )}

Remove parentheses:

(- a) = - a

= \frac{- 3 - 7}{- 2 \cdot 5}

Subtract the numbers:

- 3 - 7 = - 10

= \frac{- 10}{- 2 \cdot 5}

Multiply the numbers:

2 \cdot 5 = 10

= \frac{- 10}{- 10}

Apply the fraction rule:

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

= \frac{10}{10}

Apply rule

\frac{a}{a} = 1

= 1

The solutions to the quadratic equation are:

u = - \frac{2}{5}, u = 1

Substitute back

u = sin (x)

sin (x) = - \frac{2}{5}, sin (x) = 1

sin (x) = - \frac{2}{5}, sin (x) = 1

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

sin (x) = - \frac{2}{5}

Apply trig inverse properties

sin (x) = - \frac{2}{5}

General solutions for

sin (x) = - \frac{2}{5}

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

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

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 1

General solutions for

sin (x) = 1

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}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

Show solutions in decimal form

x = - 0.41151 \dots + 2 πn, x = π + 0.41151 \dots + 2 πn, x = \frac{π}{2} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 5cos^2(x)+3sin(x)-3=0 ?
The general solution for 5cos^2(x)+3sin(x)-3=0 is x=-0.41151…+2pin,x=pi+0.41151…+2pin,x= pi/2+2pin