What is the general solution for sin^2(x)+1/(sec(x))= 5/4 ?

The general solution for sin^2(x)+1/(sec(x))= 5/4 is x= pi/3+2pin,x=(5pi)/3+2pin

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sin^2(x)+1/(sec(x))= 5/4

Solution

sin^{2} (x) + \frac{1}{sec ( x )} = \frac{5}{4}

Solution

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

Degrees

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

Solution steps

sin^{2} (x) + \frac{1}{sec ( x )} = \frac{5}{4}

Subtract

\frac{5}{4}

from both sides

sin^{2} (x) + \frac{1}{sec ( x )} - \frac{5}{4} = 0

Simplify

sin^{2} (x) + \frac{1}{sec ( x )} - \frac{5}{4} : \frac{4 sin ^{2} ( x ) sec ( x ) + 4 - 5 sec ( x )}{4 sec ( x )}

sin^{2} (x) + \frac{1}{sec ( x )} - \frac{5}{4}

Convert element to fraction:

sin^{2} (x) = \frac{sin ^{2} ( x )}{1}

= \frac{sin ^{2} ( x )}{1} + \frac{1}{sec ( x )} - \frac{5}{4}

Least Common Multiplier of

1, sec (x), 4 : 4 sec (x)

1, sec (x), 4

Lowest Common Multiplier (LCM)

Least Common Multiplier of

1, 4 : 4

1, 4

Least Common Multiplier (LCM)

Prime factorization of

1

Prime factorization of

4 : 2 \cdot 2

4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2

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

1

4

= 2 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= 4

Compute an expression comprised of factors that appear in at least one of the factored expressions

= 4 sec (x)

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

4 sec (x)

For

\frac{sin ^{2} ( x )}{1} :

multiply the denominator and numerator by

4 sec (x)

\frac{sin ^{2} ( x )}{1} = \frac{sin ^{2} ( x ) \cdot 4 sec ( x )}{1 \cdot 4 sec ( x )} = \frac{sin ^{2} ( x ) \cdot 4 sec ( x )}{4 sec ( x )}

For

\frac{1}{sec ( x )} :

multiply the denominator and numerator by

4

\frac{1}{sec ( x )} = \frac{1 \cdot 4}{sec ( x ) \cdot 4} = \frac{4}{4 sec ( x )}

For

\frac{5}{4} :

multiply the denominator and numerator by

sec (x)

\frac{5}{4} = \frac{5 sec ( x )}{4 sec ( x )}

= \frac{sin ^{2} ( x ) \cdot 4 sec ( x )}{4 sec ( x )} + \frac{4}{4 sec ( x )} - \frac{5 sec ( x )}{4 sec ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ^{2} ( x ) \cdot 4 sec ( x ) + 4 - 5 sec ( x )}{4 sec ( x )}

\frac{4 sin ^{2} ( x ) sec ( x ) + 4 - 5 sec ( x )}{4 sec ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

4 sin^{2} (x) sec (x) + 4 - 5 sec (x) = 0

Express with sin, cos

4 sin^{2} (x) \frac{1}{cos ( x )} + 4 - 5 \cdot \frac{1}{cos ( x )} = 0

Simplify

4 sin^{2} (x) \frac{1}{cos ( x )} + 4 - 5 \cdot \frac{1}{cos ( x )} : \frac{4 sin ^{2} ( x ) - 5 + 4 cos ( x )}{cos ( x )}

4 sin^{2} (x) \frac{1}{cos ( x )} + 4 - 5 \cdot \frac{1}{cos ( x )}

4 sin^{2} (x) \frac{1}{cos ( x )} = \frac{4 sin ^{2} ( x )}{cos ( x )}

4 sin^{2} (x) \frac{1}{cos ( x )}

Multiply fractions:

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

= \frac{1 \cdot 4 sin ^{2} ( x )}{cos ( x )}

Multiply the numbers:

1 \cdot 4 = 4

= \frac{4 sin ^{2} ( x )}{cos ( x )}

5 \cdot \frac{1}{cos ( x )} = \frac{5}{cos ( x )}

5 \cdot \frac{1}{cos ( x )}

Multiply fractions:

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

= \frac{1 \cdot 5}{cos ( x )}

Multiply the numbers:

1 \cdot 5 = 5

= \frac{5}{cos ( x )}

= \frac{4 sin ^{2} ( x )}{cos ( x )} + 4 - \frac{5}{cos ( x )}

Combine the fractions

\frac{4 sin ^{2} ( x )}{cos ( x )} - \frac{5}{cos ( x )} : \frac{4 sin ^{2} ( x ) - 5}{cos ( x )}

Apply rule

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

= \frac{4 sin ^{2} ( x ) - 5}{cos ( x )}

= \frac{4 sin ^{2} ( x ) - 5}{cos ( x )} + 4

Convert element to fraction:

4 = \frac{4 cos ( x )}{cos ( x )}

= \frac{4 sin ^{2} ( x ) - 5}{cos ( x )} + \frac{4 cos ( x )}{cos ( x )}

Since the denominators are equal, combine the fractions:

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

= \frac{4 sin ^{2} ( x ) - 5 + 4 cos ( x )}{cos ( x )}

\frac{4 sin ^{2} ( x ) - 5 + 4 cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

4 sin^{2} (x) - 5 + 4 cos (x) = 0

Subtract

4 cos (x)

from both sides

4 sin^{2} (x) - 5 = - 4 cos (x)

Square both sides

(4 sin^{2} (x) - 5)^{2} = (- 4 cos (x))^{2}

Subtract

(- 4 cos (x))^{2}

from both sides

(4 sin^{2} (x) - 5)^{2} - 16 cos^{2} (x) = 0

Factor

(4 sin^{2} (x) - 5)^{2} - 16 cos^{2} (x) : (4 sin^{2} (x) - 5 + 4 cos (x)) (4 sin^{2} (x) - 5 - 4 cos (x))

(4 sin^{2} (x) - 5)^{2} - 16 cos^{2} (x)

Rewrite

(4 sin^{2} (x) - 5)^{2} - 16 cos^{2} (x)

(4 sin^{2} (x) - 5)^{2} - (4 cos (x))^{2}

(4 sin^{2} (x) - 5)^{2} - 16 cos^{2} (x)

Rewrite

16

4^{2}

= (4 sin^{2} (x) - 5)^{2} - 4^{2} cos^{2} (x)

Apply exponent rule:

a^{m} b^{m} = (ab)^{m}

4^{2} cos^{2} (x) = (4 cos (x))^{2}

= (4 sin^{2} (x) - 5)^{2} - (4 cos (x))^{2}

= (4 sin^{2} (x) - 5)^{2} - (4 cos (x))^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(4 sin^{2} (x) - 5)^{2} - (4 cos (x))^{2} = ((4 sin^{2} (x) - 5) + 4 cos (x)) ((4 sin^{2} (x) - 5) - 4 cos (x))

= ((4 sin^{2} (x) - 5) + 4 cos (x)) ((4 sin^{2} (x) - 5) - 4 cos (x))

Refine

= (4 sin^{2} (x) + 4 cos (x) - 5) (4 sin^{2} (x) - 4 cos (x) - 5)

(4 sin^{2} (x) - 5 + 4 cos (x)) (4 sin^{2} (x) - 5 - 4 cos (x)) = 0

Solving each part separately

4 sin^{2} (x) - 5 + 4 cos (x) = 0 or 4 sin^{2} (x) - 5 - 4 cos (x) = 0

4 sin^{2} (x) - 5 + 4 cos (x) = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

4 sin^{2} (x) - 5 + 4 cos (x) = 0

Rewrite using trig identities

- 5 + 4 cos (x) + 4 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 5 + 4 cos (x) + 4 (1 - cos^{2} (x))

Simplify

- 5 + 4 cos (x) + 4 (1 - cos^{2} (x)) : 4 cos (x) - 4 cos^{2} (x) - 1

- 5 + 4 cos (x) + 4 (1 - cos^{2} (x))

Expand

4 (1 - cos^{2} (x)) : 4 - 4 cos^{2} (x)

4 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (x)

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

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 cos^{2} (x)

= - 5 + 4 cos (x) + 4 - 4 cos^{2} (x)

Simplify

- 5 + 4 cos (x) + 4 - 4 cos^{2} (x) : 4 cos (x) - 4 cos^{2} (x) - 1

- 5 + 4 cos (x) + 4 - 4 cos^{2} (x)

Group like terms

= 4 cos (x) - 4 cos^{2} (x) - 5 + 4

Add/Subtract the numbers:

- 5 + 4 = - 1

= 4 cos (x) - 4 cos^{2} (x) - 1

= 4 cos (x) - 4 cos^{2} (x) - 1

= 4 cos (x) - 4 cos^{2} (x) - 1

- 1 + 4 cos (x) - 4 cos^{2} (x) = 0

Solve by substitution

- 1 + 4 cos (x) - 4 cos^{2} (x) = 0

Let:

cos (x) = u

- 1 + 4 u - 4 u^{2} = 0

- 1 + 4 u - 4 u^{2} = 0 : u = \frac{1}{2}

- 1 + 4 u - 4 u^{2} = 0

Write in the standard form

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

- 4 u^{2} + 4 u - 1 = 0

Solve with the quadratic formula

- 4 u^{2} + 4 u - 1 = 0

Quadratic Equation Formula:

For

a = - 4, b = 4, c = - 1

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

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

4^{2} - 4 (- 4) (- 1) = 0

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

Apply rule

- (- a) = a

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

Multiply the numbers:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- 4 \pm 0}{2 ( - 4 )}

u = \frac{- 4}{2 ( - 4 )}

\frac{- 4}{2 ( - 4 )} = \frac{1}{2}

\frac{- 4}{2 ( - 4 )}

Remove parentheses:

(- a) = - a

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{- 4}{- 8}

Apply the fraction rule:

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

= \frac{4}{8}

Cancel the common factor:

4

= \frac{1}{2}

u = \frac{1}{2}

The solution to the quadratic equation is:

u = \frac{1}{2}

Substitute back

u = cos (x)

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

cos (x) = \frac{1}{2}

General solutions for

cos (x) = \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}

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

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

Combine all the solutions

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

4 sin^{2} (x) - 5 - 4 cos (x) = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

4 sin^{2} (x) - 5 - 4 cos (x) = 0

Rewrite using trig identities

- 5 - 4 cos (x) + 4 sin^{2} (x)

Use the Pythagorean identity:

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

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

= - 5 - 4 cos (x) + 4 (1 - cos^{2} (x))

Simplify

- 5 - 4 cos (x) + 4 (1 - cos^{2} (x)) : - 4 cos^{2} (x) - 4 cos (x) - 1

- 5 - 4 cos (x) + 4 (1 - cos^{2} (x))

Expand

4 (1 - cos^{2} (x)) : 4 - 4 cos^{2} (x)

4 (1 - cos^{2} (x))

Apply the distributive law:

a (b - c) = ab - a c

a = 4, b = 1, c = cos^{2} (x)

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

Multiply the numbers:

4 \cdot 1 = 4

= 4 - 4 cos^{2} (x)

= - 5 - 4 cos (x) + 4 - 4 cos^{2} (x)

Simplify

- 5 - 4 cos (x) + 4 - 4 cos^{2} (x) : - 4 cos^{2} (x) - 4 cos (x) - 1

- 5 - 4 cos (x) + 4 - 4 cos^{2} (x)

Group like terms

= - 4 cos (x) - 4 cos^{2} (x) - 5 + 4

Add/Subtract the numbers:

- 5 + 4 = - 1

= - 4 cos^{2} (x) - 4 cos (x) - 1

= - 4 cos^{2} (x) - 4 cos (x) - 1

= - 4 cos^{2} (x) - 4 cos (x) - 1

- 1 - 4 cos (x) - 4 cos^{2} (x) = 0

Solve by substitution

- 1 - 4 cos (x) - 4 cos^{2} (x) = 0

Let:

cos (x) = u

- 1 - 4 u - 4 u^{2} = 0

- 1 - 4 u - 4 u^{2} = 0 : u = - \frac{1}{2}

- 1 - 4 u - 4 u^{2} = 0

Write in the standard form

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

- 4 u^{2} - 4 u - 1 = 0

Solve with the quadratic formula

- 4 u^{2} - 4 u - 1 = 0

Quadratic Equation Formula:

For

a = - 4, b = - 4, c = - 1

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

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

(- 4)^{2} - 4 (- 4) (- 1) = 0

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

Apply rule

- (- a) = a

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

Apply exponent rule:

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

n

is even

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

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

Multiply the numbers:

4 \cdot 4 \cdot 1 = 16

= 4^{2} - 16

4^{2} = 16

= 16 - 16

Subtract the numbers:

16 - 16 = 0

= 0

u_{1, 2} = \frac{- ( - 4 ) \pm 0}{2 ( - 4 )}

u = \frac{- ( - 4 )}{2 ( - 4 )}

\frac{- ( - 4 )}{2 ( - 4 )} = - \frac{1}{2}

\frac{- ( - 4 )}{2 ( - 4 )}

Remove parentheses:

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

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{4}{- 8}

Apply the fraction rule:

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

= - \frac{4}{8}

Cancel the common factor:

4

= - \frac{1}{2}

u = - \frac{1}{2}

The solution to the quadratic equation is:

u = - \frac{1}{2}

Substitute back

u = cos (x)

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

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

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

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

General solutions for

cos (x) = - \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}

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

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

Combine all the solutions

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

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Verify solutions by plugging them into the original equation

Check the solutions by plugging them into

sin^{2} (x) + \frac{1}{sec ( x )} = \frac{5}{4}

Remove the ones that don't agree with the equation.

Check the solution

\frac{π}{3} + 2 πn :

True

\frac{π}{3} + 2 πn

Plug in

n = 1

\frac{π}{3} + 2 π 1

For

sin^{2} (x) + \frac{1}{sec ( x )} = \frac{5}{4}

plug in

x = \frac{π}{3} + 2 π 1

sin^{2} (\frac{π}{3} + 2 π 1) + \frac{1}{sec ( \frac{π}{3} + 2 π 1 )} = \frac{5}{4}

Refine

1.25 = 1.25

\Rightarrow True

Check the solution

\frac{5 π}{3} + 2 πn :

True

\frac{5 π}{3} + 2 πn

Plug in

n = 1

\frac{5 π}{3} + 2 π 1

For

sin^{2} (x) + \frac{1}{sec ( x )} = \frac{5}{4}

plug in

x = \frac{5 π}{3} + 2 π 1

sin^{2} (\frac{5 π}{3} + 2 π 1) + \frac{1}{sec ( \frac{5 π}{3} + 2 π 1 )} = \frac{5}{4}

Refine

1.25 = 1.25

\Rightarrow True

Check the solution

\frac{2 π}{3} + 2 πn :

False

\frac{2 π}{3} + 2 πn

Plug in

n = 1

\frac{2 π}{3} + 2 π 1

For

sin^{2} (x) + \frac{1}{sec ( x )} = \frac{5}{4}

plug in

x = \frac{2 π}{3} + 2 π 1

sin^{2} (\frac{2 π}{3} + 2 π 1) + \frac{1}{sec ( \frac{2 π}{3} + 2 π 1 )} = \frac{5}{4}

Refine

0.25 = 1.25

\Rightarrow False

Check the solution

\frac{4 π}{3} + 2 πn :

False

\frac{4 π}{3} + 2 πn

Plug in

n = 1

\frac{4 π}{3} + 2 π 1

For

sin^{2} (x) + \frac{1}{sec ( x )} = \frac{5}{4}

plug in

x = \frac{4 π}{3} + 2 π 1

sin^{2} (\frac{4 π}{3} + 2 π 1) + \frac{1}{sec ( \frac{4 π}{3} + 2 π 1 )} = \frac{5}{4}

Refine

0.25 = 1.25

\Rightarrow False

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

Popular Examples

2cos(1/5 θ)=sqrt(3)

2 cos (\frac{1}{5} θ) = 3

sec(θ)=-7/(sqrt(13))

sec (θ) = - \frac{7}{13}

tan(x)=-infinity

tan (x) = - \infty

tan(2x)=-tan(x)2

tan (2 x) = - tan (x) 2

2sin^2(x)-5sin(x)-12=0

2 sin^{2} (x) - 5 sin (x) - 12 = 0

Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)+1/(sec(x))= 5/4 ?
The general solution for sin^2(x)+1/(sec(x))= 5/4 is x= pi/3+2pin,x=(5pi)/3+2pin