What is the general solution for 3tan^2(y)=5sec(y)-1 ?

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

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3tan^2(y)=5sec(y)-1

Solution

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

Solution

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

Degrees

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

Solution steps

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

Subtract

5 sec (y) - 1

from both sides

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

Rewrite using trig identities

1 + 3 tan^{2} (y) - 5 sec (y)

Use the Pythagorean identity:

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

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

= 1 + 3 (sec^{2} (y) - 1) - 5 sec (y)

Simplify

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

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

Expand

3 (sec^{2} (y) - 1) : 3 sec^{2} (y) - 3

3 (sec^{2} (y) - 1)

Apply the distributive law:

a (b - c) = ab - a c

a = 3, b = sec^{2} (y), c = 1

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

Multiply the numbers:

3 \cdot 1 = 3

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

= 1 + 3 sec^{2} (y) - 3 - 5 sec (y)

Simplify

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

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

Group like terms

= 3 sec^{2} (y) - 5 sec (y) + 1 - 3

Add/Subtract the numbers:

1 - 3 = - 2

= 3 sec^{2} (y) - 5 sec (y) - 2

= 3 sec^{2} (y) - 5 sec (y) - 2

= 3 sec^{2} (y) - 5 sec (y) - 2

- 2 + 3 sec^{2} (y) - 5 sec (y) = 0

Solve by substitution

- 2 + 3 sec^{2} (y) - 5 sec (y) = 0

Let:

sec (y) = u

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

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

- 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 \cdot 3 ( - 2 )}{2 \cdot 3}

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

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

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

= 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:

7^{2} = 7

= 7

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

Separate the solutions

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

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

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

Apply rule

- (- a) = a

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

Add the numbers:

5 + 7 = 12

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

Multiply the numbers:

2 \cdot 3 = 6

= \frac{12}{6}

Divide the numbers:

\frac{12}{6} = 2

= 2

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

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

Apply rule

- (- a) = a

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

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}

The solutions to the quadratic equation are:

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

Substitute back

u = sec (y)

sec (y) = 2, sec (y) = - \frac{1}{3}

sec (y) = 2, sec (y) = - \frac{1}{3}

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

sec (y) = 2

General solutions for

sec (y) = 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}

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

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

sec (y) = - \frac{1}{3} :

No Solution

sec (y) = - \frac{1}{3}

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

No Solution

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

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