What is the general solution for sec(y+pi/2)=-2 ?

The general solution for sec(y+pi/2)=-2 is y=2pin+pi/6 ,y=2pin+(5pi)/6

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Popular Trigonometry >

sec(y+pi/2)=-2

Solution

sec (y + \frac{π}{2}) = - 2

Solution

y = 2 πn + \frac{π}{6}, y = 2 πn + \frac{5 π}{6}

Degrees

y = 3 0^{\circ} + 36 0^{\circ} n, y = 15 0^{\circ} + 36 0^{\circ} n

Solution steps

sec (y + \frac{π}{2}) = - 2

General solutions for

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

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

Solve

y + \frac{π}{2} = \frac{2 π}{3} + 2 πn : y = 2 πn + \frac{π}{6}

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

Move

\frac{π}{2}

to the right side

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

Subtract

\frac{π}{2}

from both sides

y + \frac{π}{2} - \frac{π}{2} = \frac{2 π}{3} + 2 πn - \frac{π}{2}

Simplify

y + \frac{π}{2} - \frac{π}{2} = \frac{2 π}{3} + 2 πn - \frac{π}{2}

Simplify

y + \frac{π}{2} - \frac{π}{2} : y

y + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= y

Simplify

\frac{2 π}{3} + 2 πn - \frac{π}{2} : 2 πn + \frac{π}{6}

\frac{2 π}{3} + 2 πn - \frac{π}{2}

Group like terms

= 2 πn - \frac{π}{2} + \frac{2 π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{2 π}{3} :

multiply the denominator and numerator by

2

\frac{2 π}{3} = \frac{2 π 2}{3 \cdot 2} = \frac{4 π}{6}

= - \frac{π 3}{6} + \frac{4 π}{6}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- π 3 + 4 π}{6}

Add similar elements:

- 3 π + 4 π = π

= 2 πn + \frac{π}{6}

y = 2 πn + \frac{π}{6}

y = 2 πn + \frac{π}{6}

y = 2 πn + \frac{π}{6}

Solve

y + \frac{π}{2} = \frac{4 π}{3} + 2 πn : y = 2 πn + \frac{5 π}{6}

y + \frac{π}{2} = \frac{4 π}{3} + 2 πn

Move

\frac{π}{2}

to the right side

y + \frac{π}{2} = \frac{4 π}{3} + 2 πn

Subtract

\frac{π}{2}

from both sides

y + \frac{π}{2} - \frac{π}{2} = \frac{4 π}{3} + 2 πn - \frac{π}{2}

Simplify

y + \frac{π}{2} - \frac{π}{2} = \frac{4 π}{3} + 2 πn - \frac{π}{2}

Simplify

y + \frac{π}{2} - \frac{π}{2} : y

y + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= y

Simplify

\frac{4 π}{3} + 2 πn - \frac{π}{2} : 2 πn + \frac{5 π}{6}

\frac{4 π}{3} + 2 πn - \frac{π}{2}

Group like terms

= 2 πn - \frac{π}{2} + \frac{4 π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{4 π}{3} :

multiply the denominator and numerator by

2

\frac{4 π}{3} = \frac{4 π 2}{3 \cdot 2} = \frac{8 π}{6}

= - \frac{π 3}{6} + \frac{8 π}{6}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- π 3 + 8 π}{6}

Add similar elements:

- 3 π + 8 π = 5 π

= 2 πn + \frac{5 π}{6}

y = 2 πn + \frac{5 π}{6}

y = 2 πn + \frac{5 π}{6}

y = 2 πn + \frac{5 π}{6}

y = 2 πn + \frac{π}{6}, y = 2 πn + \frac{5 π}{6}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sec(y+pi/2)=-2 ?
The general solution for sec(y+pi/2)=-2 is y=2pin+pi/6 ,y=2pin+(5pi)/6