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Popular Trigonometry >

sin^2(x+pi/2)= 3/4

Solution

sin^{2} (x + \frac{π}{2}) = \frac{3}{4}

Solution

x = 2 πn - \frac{π}{6}, x = 2 πn + \frac{π}{6}, x = 2 πn + \frac{5 π}{6}, x = 2 πn + \frac{7 π}{6}

Degrees

x = - 3 0^{\circ} + 36 0^{\circ} n, x = 3 0^{\circ} + 36 0^{\circ} n, x = 15 0^{\circ} + 36 0^{\circ} n, x = 21 0^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x + \frac{π}{2}) = \frac{3}{4}

Solve by substitution

sin^{2} (x + \frac{π}{2}) = \frac{3}{4}

Let:

sin (x + \frac{π}{2}) = u

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4} : u = \frac{3}{2}, u = - \frac{3}{2}

u^{2} = \frac{3}{4}

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Simplify

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

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

Substitute back

u = sin (x + \frac{π}{2})

sin (x + \frac{π}{2}) = \frac{3}{2}, sin (x + \frac{π}{2}) = - \frac{3}{2}

sin (x + \frac{π}{2}) = \frac{3}{2}, sin (x + \frac{π}{2}) = - \frac{3}{2}

sin (x + \frac{π}{2}) = \frac{3}{2} : x = 2 πn - \frac{π}{6}, x = 2 πn + \frac{π}{6}

sin (x + \frac{π}{2}) = \frac{3}{2}

General solutions for

sin (x + \frac{π}{2}) = \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Solve

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

x + \frac{π}{2} = \frac{π}{3} + 2 πn

Move

\frac{π}{2}

to the right side

x + \frac{π}{2} = \frac{π}{3} + 2 πn

Subtract

\frac{π}{2}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{2} - \frac{π}{2} : x

x + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= x

Simplify

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

\frac{π}{3} + 2 πn - \frac{π}{2}

Group like terms

= 2 πn + \frac{π}{3} - \frac{π}{2}

Least Common Multiplier of

3, 2 : 6

3, 2

Least Common Multiplier (LCM)

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

3

2

= 3 \cdot 2

Multiply the numbers:

3 \cdot 2 = 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}{3} :

multiply the denominator and numerator by

2

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

= \frac{π 2}{6} - \frac{π 3}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π 3}{6}

Add similar elements:

2 π - 3 π = - π

= \frac{- π}{6}

Apply the fraction rule:

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

= 2 πn - \frac{π}{6}

x = 2 πn - \frac{π}{6}

x = 2 πn - \frac{π}{6}

x = 2 πn - \frac{π}{6}

Solve

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

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

Move

\frac{π}{2}

to the right side

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

Subtract

\frac{π}{2}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{2} - \frac{π}{2} : x

x + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= x

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}

x = 2 πn + \frac{π}{6}

x = 2 πn + \frac{π}{6}

x = 2 πn + \frac{π}{6}

x = 2 πn - \frac{π}{6}, x = 2 πn + \frac{π}{6}

sin (x + \frac{π}{2}) = - \frac{3}{2} : x = 2 πn + \frac{5 π}{6}, x = 2 πn + \frac{7 π}{6}

sin (x + \frac{π}{2}) = - \frac{3}{2}

General solutions for

sin (x + \frac{π}{2}) = - \frac{3}{2}

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

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

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

Solve

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

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

Move

\frac{π}{2}

to the right side

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

Subtract

\frac{π}{2}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{2} - \frac{π}{2} : x

x + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= x

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}

x = 2 πn + \frac{5 π}{6}

x = 2 πn + \frac{5 π}{6}

x = 2 πn + \frac{5 π}{6}

Solve

x + \frac{π}{2} = \frac{5 π}{3} + 2 πn : x = 2 πn + \frac{7 π}{6}

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

Move

\frac{π}{2}

to the right side

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

Subtract

\frac{π}{2}

from both sides

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

Simplify

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

Simplify

x + \frac{π}{2} - \frac{π}{2} : x

x + \frac{π}{2} - \frac{π}{2}

Add similar elements:

\frac{π}{2} - \frac{π}{2} = 0

= x

Simplify

\frac{5 π}{3} + 2 πn - \frac{π}{2} : 2 πn + \frac{7 π}{6}

\frac{5 π}{3} + 2 πn - \frac{π}{2}

Group like terms

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

multiply the denominator and numerator by

2

\frac{5 π}{3} = \frac{5 π 2}{3 \cdot 2} = \frac{10 π}{6}

= - \frac{π 3}{6} + \frac{10 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 + 10 π}{6}

Add similar elements:

- 3 π + 10 π = 7 π

= 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{7 π}{6}

x = 2 πn + \frac{5 π}{6}, x = 2 πn + \frac{7 π}{6}

Combine all the solutions

x = 2 πn - \frac{π}{6}, x = 2 πn + \frac{π}{6}, x = 2 πn + \frac{5 π}{6}, x = 2 πn + \frac{7 π}{6}

Graph

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