What is the general solution for 2sin^2(t)-3cos(t)=3 ?

The general solution for 2sin^2(t)-3cos(t)=3 is t=pi+2pin,t=(2pi)/3+2pin,t=(4pi)/3+2pin

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2sin^2(t)-3cos(t)=3

Solution

2 sin^{2} (t) - 3 cos (t) = 3

Solution

t = π + 2 πn, t = \frac{2 π}{3} + 2 πn, t = \frac{4 π}{3} + 2 πn

Degrees

t = 18 0^{\circ} + 36 0^{\circ} n, t = 12 0^{\circ} + 36 0^{\circ} n, t = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sin^{2} (t) - 3 cos (t) = 3

Subtract

3

from both sides

2 sin^{2} (t) - 3 cos (t) - 3 = 0

Rewrite using trig identities

- 3 + 2 sin^{2} (t) - 3 cos (t)

Use the Pythagorean identity:

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

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

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

Simplify

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

- 3 + 2 (1 - cos^{2} (t)) - 3 cos (t)

Expand

2 (1 - cos^{2} (t)) : 2 - 2 cos^{2} (t)

2 (1 - cos^{2} (t))

Apply the distributive law:

a (b - c) = ab - a c

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

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

Multiply the numbers:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (t)

= - 3 + 2 - 2 cos^{2} (t) - 3 cos (t)

Add/Subtract the numbers:

- 3 + 2 = - 1

= - 2 cos^{2} (t) - 3 cos (t) - 1

= - 2 cos^{2} (t) - 3 cos (t) - 1

- 1 - 2 cos^{2} (t) - 3 cos (t) = 0

Solve by substitution

- 1 - 2 cos^{2} (t) - 3 cos (t) = 0

Let:

cos (t) = u

- 1 - 2 u^{2} - 3 u = 0

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

- 1 - 2 u^{2} - 3 u = 0

Write in the standard form

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

- 2 u^{2} - 3 u - 1 = 0

Solve with the quadratic formula

- 2 u^{2} - 3 u - 1 = 0

Quadratic Equation Formula:

For

a = - 2, b = - 3, c = - 1

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

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

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

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

Apply rule

- (- a) = a

= (- 3)^{2} - 4 \cdot 2 \cdot 1

Apply exponent rule:

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

n

is even

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

= 3^{2} - 4 \cdot 2 \cdot 1

Multiply the numbers:

4 \cdot 2 \cdot 1 = 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

Subtract the numbers:

9 - 8 = 1

= 1

Apply rule

1 = 1

= 1

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

Separate the solutions

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

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

\frac{- ( - 3 ) + 1}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 + 1}{- 2 \cdot 2}

Add the numbers:

3 + 1 = 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{- ( - 3 ) - 1}{2 ( - 2 )} : - \frac{1}{2}

\frac{- ( - 3 ) - 1}{2 ( - 2 )}

Remove parentheses:

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

= \frac{3 - 1}{- 2 \cdot 2}

Subtract the numbers:

3 - 1 = 2

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2}{- 4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

The solutions to the quadratic equation are:

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

Substitute back

u = cos (t)

cos (t) = - 1, cos (t) = - \frac{1}{2}

cos (t) = - 1, cos (t) = - \frac{1}{2}

cos (t) = - 1 : t = π + 2 πn

cos (t) = - 1

General solutions for

cos (t) = - 1

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

t = π + 2 πn

t = π + 2 πn

cos (t) = - \frac{1}{2} : t = \frac{2 π}{3} + 2 πn, t = \frac{4 π}{3} + 2 πn

cos (t) = - \frac{1}{2}

General solutions for

cos (t) = - \frac{1}{2}

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

t = \frac{2 π}{3} + 2 πn, t = \frac{4 π}{3} + 2 πn

t = \frac{2 π}{3} + 2 πn, t = \frac{4 π}{3} + 2 πn

Combine all the solutions

t = π + 2 πn, t = \frac{2 π}{3} + 2 πn, t = \frac{4 π}{3} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin^2(t)-3cos(t)=3 ?
The general solution for 2sin^2(t)-3cos(t)=3 is t=pi+2pin,t=(2pi)/3+2pin,t=(4pi)/3+2pin