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Popular Trigonometry >

sin^2(x)+5/14 = 6/7

Solution

sin^{2} (x) + \frac{5}{14} = \frac{6}{7}

Solution

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Degrees

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x) + \frac{5}{14} = \frac{6}{7}

Solve by substitution

sin^{2} (x) + \frac{5}{14} = \frac{6}{7}

Let:

sin (x) = u

u^{2} + \frac{5}{14} = \frac{6}{7}

u^{2} + \frac{5}{14} = \frac{6}{7} : u = \frac{1}{2}, u = - \frac{1}{2}

u^{2} + \frac{5}{14} = \frac{6}{7}

Move

\frac{5}{14}

to the right side

u^{2} + \frac{5}{14} = \frac{6}{7}

Subtract

\frac{5}{14}

from both sides

u^{2} + \frac{5}{14} - \frac{5}{14} = \frac{6}{7} - \frac{5}{14}

Simplify

u^{2} = \frac{6}{7} - \frac{5}{14}

u^{2} = \frac{6}{7} - \frac{5}{14}

Simplify

\frac{6}{7} - \frac{5}{14} : \frac{1}{2}

\frac{6}{7} - \frac{5}{14}

Least Common Multiplier of

7, 14 : 14

7, 14

Least Common Multiplier (LCM)

Prime factorization of

7 : 7

7

7

is a prime number, therefore no factorization is possible

= 7

Prime factorization of

14 : 2 \cdot 7

14

14

divides by

214 = 7 \cdot 2

= 2 \cdot 7

2, 7

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 7

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

7

14

= 7 \cdot 2

Multiply the numbers:

7 \cdot 2 = 14

= 14

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

14

For

\frac{6}{7} :

multiply the denominator and numerator by

2

\frac{6}{7} = \frac{6 \cdot 2}{7 \cdot 2} = \frac{12}{14}

= \frac{12}{14} - \frac{5}{14}

Since the denominators are equal, combine the fractions:

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

= \frac{12 - 5}{14}

Subtract the numbers:

12 - 5 = 7

= \frac{7}{14}

Cancel the common factor:

7

= \frac{1}{2}

For

x^{2} = f (a)

the solutions are

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

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

Substitute back

u = sin (x)

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

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

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

General solutions for

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

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

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

General solutions for

sin (x) = - \frac{1}{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{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Graph

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