What is the general solution for sec^2(t)+2sec(t)=0 ?

The general solution for sec^2(t)+2sec(t)=0 is t=(2pi)/3+2pin,t=(4pi)/3+2pin

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Popular >

sec^2(t)+2sec(t)=0

Solution

sec^{2} (t) + 2 sec (t) = 0

Solution

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

Degrees

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

Solution steps

sec^{2} (t) + 2 sec (t) = 0

Solve by substitution

sec^{2} (t) + 2 sec (t) = 0

Let:

sec (t) = u

u^{2} + 2 u = 0

u^{2} + 2 u = 0 : u = 0, u = - 2

u^{2} + 2 u = 0

Solve with the quadratic formula

u^{2} + 2 u = 0

Quadratic Equation Formula:

For

a = 1, b = 2, c = 0

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

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

2^{2} - 4 \cdot 1 \cdot 0 = 2

2^{2} - 4 \cdot 1 \cdot 0

Apply rule

0 \cdot a = 0

= 2^{2} - 0

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

= 2^{2}

Apply radical rule:

assuming

a \geq 0

= 2

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

Separate the solutions

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

u = \frac{- 2 + 2}{2 \cdot 1} : 0

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

Add/Subtract the numbers:

- 2 + 2 = 0

= \frac{0}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{0}{2}

Apply rule

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

= 0

u = \frac{- 2 - 2}{2 \cdot 1} : - 2

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

Subtract the numbers:

- 2 - 2 = - 4

= \frac{- 4}{2 \cdot 1}

Multiply the numbers:

2 \cdot 1 = 2

= \frac{- 4}{2}

Apply the fraction rule:

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

= - \frac{4}{2}

Divide the numbers:

\frac{4}{2} = 2

= - 2

The solutions to the quadratic equation are:

u = 0, u = - 2

Substitute back

u = sec (t)

sec (t) = 0, sec (t) = - 2

sec (t) = 0, sec (t) = - 2

sec (t) = 0 :

No Solution

sec (t) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

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

sec (t) = - 2

General solutions for

sec (t) = - 2

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

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 = \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 sec^2(t)+2sec(t)=0 ?
The general solution for sec^2(t)+2sec(t)=0 is t=(2pi)/3+2pin,t=(4pi)/3+2pin