What is the general solution for 2cos^2(x)+15sin(x)-15=0 ?

The general solution for 2cos^2(x)+15sin(x)-15=0 is x= pi/2+2pin

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Popular >

2cos^2(x)+15sin(x)-15=0

Solution

2 cos^{2} (x) + 15 sin (x) - 15 = 0

Solution

x = \frac{π}{2} + 2 πn

Degrees

x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

2 cos^{2} (x) + 15 sin (x) - 15 = 0

Rewrite using trig identities

- 15 + 15 sin (x) + 2 cos^{2} (x)

Use the Pythagorean identity:

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

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

= - 15 + 15 sin (x) + 2 (1 - sin^{2} (x))

Simplify

- 15 + 15 sin (x) + 2 (1 - sin^{2} (x)) : 15 sin (x) - 2 sin^{2} (x) - 13

- 15 + 15 sin (x) + 2 (1 - sin^{2} (x))

Expand

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

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

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

2 \cdot 1 = 2

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

= - 15 + 15 sin (x) + 2 - 2 sin^{2} (x)

Simplify

- 15 + 15 sin (x) + 2 - 2 sin^{2} (x) : 15 sin (x) - 2 sin^{2} (x) - 13

- 15 + 15 sin (x) + 2 - 2 sin^{2} (x)

Group like terms

= 15 sin (x) - 2 sin^{2} (x) - 15 + 2

Add/Subtract the numbers:

- 15 + 2 = - 13

= 15 sin (x) - 2 sin^{2} (x) - 13

= 15 sin (x) - 2 sin^{2} (x) - 13

= 15 sin (x) - 2 sin^{2} (x) - 13

- 13 + 15 sin (x) - 2 sin^{2} (x) = 0

Solve by substitution

- 13 + 15 sin (x) - 2 sin^{2} (x) = 0

Let:

sin (x) = u

- 13 + 15 u - 2 u^{2} = 0

- 13 + 15 u - 2 u^{2} = 0 : u = 1, u = \frac{13}{2}

- 13 + 15 u - 2 u^{2} = 0

Write in the standard form

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

- 2 u^{2} + 15 u - 13 = 0

Solve with the quadratic formula

- 2 u^{2} + 15 u - 13 = 0

Quadratic Equation Formula:

For

a = - 2, b = 15, c = - 13

u_{1, 2} = \frac{- 15 \pm 1 5 ^{2} - 4 ( - 2 ) ( - 13 )}{2 ( - 2 )}

u_{1, 2} = \frac{- 15 \pm 1 5 ^{2} - 4 ( - 2 ) ( - 13 )}{2 ( - 2 )}

1 5^{2} - 4 (- 2) (- 13) = 11

1 5^{2} - 4 (- 2) (- 13)

Apply rule

- (- a) = a

= 1 5^{2} - 4 \cdot 2 \cdot 13

Multiply the numbers:

4 \cdot 2 \cdot 13 = 104

= 1 5^{2} - 104

1 5^{2} = 225

= 225 - 104

Subtract the numbers:

225 - 104 = 121

= 121

Factor the number:

121 = 1 1^{2}

= 1 1^{2}

Apply radical rule:

1 1^{2} = 11

= 11

u_{1, 2} = \frac{- 15 \pm 11}{2 ( - 2 )}

Separate the solutions

u_{1} = \frac{- 15 + 11}{2 ( - 2 )}, u_{2} = \frac{- 15 - 11}{2 ( - 2 )}

u = \frac{- 15 + 11}{2 ( - 2 )} : 1

\frac{- 15 + 11}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 15 + 11}{- 2 \cdot 2}

Add/Subtract the numbers:

- 15 + 11 = - 4

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4}{- 4}

Apply the fraction rule:

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

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

u = \frac{- 15 - 11}{2 ( - 2 )} : \frac{13}{2}

\frac{- 15 - 11}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- 15 - 11}{- 2 \cdot 2}

Subtract the numbers:

- 15 - 11 = - 26

= \frac{- 26}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 26}{- 4}

Apply the fraction rule:

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

= \frac{26}{4}

Cancel the common factor:

2

= \frac{13}{2}

The solutions to the quadratic equation are:

u = 1, u = \frac{13}{2}

Substitute back

u = sin (x)

sin (x) = 1, sin (x) = \frac{13}{2}

sin (x) = 1, sin (x) = \frac{13}{2}

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

sin (x) = \frac{13}{2} :

No Solution

sin (x) = \frac{13}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{2} + 2 πn

Popular Examples

cos(2x)=sin(x),-2pi<= x<= 2pi

cos (2 x) = sin (x), - 2 π \leq x \leq 2 π

5sec^2(θ)sin(θ)-cos(θ)=0

5 sec^{2} (θ) sin (θ) - cos (θ) = 0

cos(x)= 8/11

cos (x) = \frac{8}{11}

4sec(θ)-sqrt(3)=sqrt(3)+7sec(θ)

4 sec (θ) - 3 = 3 + 7 sec (θ)

1/(sqrt(15))=cos(x)

\frac{1}{15} = cos (x)

Frequently Asked Questions (FAQ)

What is the general solution for 2cos^2(x)+15sin(x)-15=0 ?
The general solution for 2cos^2(x)+15sin(x)-15=0 is x= pi/2+2pin