What is the general solution for 5sec(θ)-2sec^2(θ)=tan^2(θ)-1 ?

The general solution for 5sec(θ)-2sec^2(θ)=tan^2(θ)-1 is θ= pi/3+2pin,θ=(5pi)/3+2pin

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Popular >

5sec(θ)-2sec^2(θ)=tan^2(θ)-1

Solution

5 sec (θ) - 2 sec^{2} (θ) = tan^{2} (θ) - 1

Solution

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

5 sec (θ) - 2 sec^{2} (θ) = tan^{2} (θ) - 1

Subtract

tan^{2} (θ) - 1

from both sides

5 sec (θ) - 2 sec^{2} (θ) - tan^{2} (θ) + 1 = 0

Rewrite using trig identities

1 - tan^{2} (θ) - 2 sec^{2} (θ) + 5 sec (θ)

Use the Pythagorean identity:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= 1 - (sec^{2} (θ) - 1) - 2 sec^{2} (θ) + 5 sec (θ)

Simplify

1 - (sec^{2} (θ) - 1) - 2 sec^{2} (θ) + 5 sec (θ) : 5 sec (θ) - 3 sec^{2} (θ) + 2

1 - (sec^{2} (θ) - 1) - 2 sec^{2} (θ) + 5 sec (θ)

- (sec^{2} (θ) - 1) : - sec^{2} (θ) + 1

- (sec^{2} (θ) - 1)

Distribute parentheses

= - (sec^{2} (θ)) - (- 1)

Apply minus-plus rules

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

= - sec^{2} (θ) + 1

= 1 - sec^{2} (θ) + 1 - 2 sec^{2} (θ) + 5 sec (θ)

Simplify

1 - sec^{2} (θ) + 1 - 2 sec^{2} (θ) + 5 sec (θ) : 5 sec (θ) - 3 sec^{2} (θ) + 2

1 - sec^{2} (θ) + 1 - 2 sec^{2} (θ) + 5 sec (θ)

Group like terms

= - sec^{2} (θ) - 2 sec^{2} (θ) + 5 sec (θ) + 1 + 1

Add similar elements:

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

= - 3 sec^{2} (θ) + 5 sec (θ) + 1 + 1

Add the numbers:

1 + 1 = 2

= 5 sec (θ) - 3 sec^{2} (θ) + 2

= 5 sec (θ) - 3 sec^{2} (θ) + 2

= 5 sec (θ) - 3 sec^{2} (θ) + 2

2 - 3 sec^{2} (θ) + 5 sec (θ) = 0

Solve by substitution

2 - 3 sec^{2} (θ) + 5 sec (θ) = 0

Let:

sec (θ) = u

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

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

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

Write in the standard form

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

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

Solve with the quadratic formula

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

Quadratic Equation Formula:

For

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

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

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

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

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 3 \cdot 2 = 24

= 5^{2} + 24

5^{2} = 25

= 25 + 24

Add the numbers:

25 + 24 = 49

= 49

Factor the number:

49 = 7^{2}

= 7^{2}

Apply radical rule:

n a^{n} = a

7^{2} = 7

= 7

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

Separate the solutions

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

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

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

Remove parentheses:

(- a) = - a

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

Add/Subtract the numbers:

- 5 + 7 = 2

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{2}{- 6}

Apply the fraction rule:

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

= - \frac{2}{6}

Cancel the common factor:

2

= - \frac{1}{3}

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

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

Remove parentheses:

(- a) = - a

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

Subtract the numbers:

- 5 - 7 = - 12

= \frac{- 12}{- 2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{- 12}{- 6}

Apply the fraction rule:

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

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

The solutions to the quadratic equation are:

u = - \frac{1}{3}, u = 2

Substitute back

u = sec (θ)

sec (θ) = - \frac{1}{3}, sec (θ) = 2

sec (θ) = - \frac{1}{3}, sec (θ) = 2

sec (θ) = - \frac{1}{3} :

No Solution

sec (θ) = - \frac{1}{3}

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

No Solution

sec (θ) = 2 : θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

sec (θ) = 2

General solutions for

sec (θ) = 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}

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Combine all the solutions

θ = \frac{π}{3} + 2 πn, θ = \frac{5 π}{3} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 5sec(θ)-2sec^2(θ)=tan^2(θ)-1 ?
The general solution for 5sec(θ)-2sec^2(θ)=tan^2(θ)-1 is θ= pi/3+2pin,θ=(5pi)/3+2pin