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Popular >

sin^2(x)= 45/52

Solution

sin^{2} (x) = \frac{45}{52}

Solution

x = 1.19512 \dots + 2 πn, x = π - 1.19512 \dots + 2 πn, x = - 1.19512 \dots + 2 πn, x = π + 1.19512 \dots + 2 πn

Degrees

x = 68.47546 \dots^{\circ} + 36 0^{\circ} n, x = 111.52453 \dots^{\circ} + 36 0^{\circ} n, x = - 68.47546 \dots^{\circ} + 36 0^{\circ} n, x = 248.47546 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (x) = \frac{45}{52}

Solve by substitution

sin^{2} (x) = \frac{45}{52}

Let:

sin (x) = u

u^{2} = \frac{45}{52}

u^{2} = \frac{45}{52} : u = \frac{3 65}{26}, u = - \frac{3 65}{26}

u^{2} = \frac{45}{52}

For

x^{2} = f (a)

the solutions are

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

u = \frac{45}{52}, u = - \frac{45}{52}

\frac{45}{52} = \frac{3 65}{26}

\frac{45}{52}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{45}{52}

52 = 213

52

Prime factorization of

52 : 2^{2} \cdot 13

52

52

divides by

252 = 26 \cdot 2

= 2 \cdot 26

26

divides by

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

Apply radical rule:

= 13 2^{2}

Apply radical rule:

2^{2} = 2

= 213

= \frac{45}{2 13}

45 = 35

45

Prime factorization of

45 : 3^{2} \cdot 5

45

45

divides by

345 = 15 \cdot 3

= 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 5

= 3^{2} \cdot 5

= 3^{2} \cdot 5

Apply radical rule:

= 5 3^{2}

Apply radical rule:

3^{2} = 3

= 35

= \frac{3 5}{2 13}

Rationalize

\frac{3 5}{2 13} : \frac{3 65}{26}

\frac{3 5}{2 13}

Multiply by the conjugate

\frac{13}{13}

= \frac{3 5 13}{2 13 13}

3513 = 365

3513

Apply radical rule:

a b = a \cdot b

513 = 5 \cdot 13

= 3 5 \cdot 13

Multiply the numbers:

5 \cdot 13 = 65

= 365

21313 = 26

21313

Apply radical rule:

a a = a

1313 = 13

= 2 \cdot 13

Multiply the numbers:

2 \cdot 13 = 26

= 26

= \frac{3 65}{26}

= \frac{3 65}{26}

- \frac{45}{52} = - \frac{3 65}{26}

- \frac{45}{52}

Simplify

\frac{45}{52} : \frac{3 5}{2 13}

\frac{45}{52}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{45}{52}

52 = 213

52

Prime factorization of

52 : 2^{2} \cdot 13

52

52

divides by

252 = 26 \cdot 2

= 2 \cdot 26

26

divides by

226 = 13 \cdot 2

= 2 \cdot 2 \cdot 13

2, 13

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 13

= 2^{2} \cdot 13

= 2^{2} \cdot 13

Apply radical rule:

= 13 2^{2}

Apply radical rule:

2^{2} = 2

= 213

= \frac{45}{2 13}

45 = 35

45

Prime factorization of

45 : 3^{2} \cdot 5

45

45

divides by

345 = 15 \cdot 3

= 3 \cdot 15

15

divides by

315 = 5 \cdot 3

= 3 \cdot 3 \cdot 5

3, 5

are all prime numbers, therefore no further factorization is possible

= 3 \cdot 3 \cdot 5

= 3^{2} \cdot 5

= 3^{2} \cdot 5

Apply radical rule:

= 5 3^{2}

Apply radical rule:

3^{2} = 3

= 35

= \frac{3 5}{2 13}

= - \frac{3 5}{2 13}

Rationalize

- \frac{3 5}{2 13} : - \frac{3 65}{26}

- \frac{3 5}{2 13}

Multiply by the conjugate

\frac{13}{13}

= - \frac{3 5 13}{2 13 13}

3513 = 365

3513

Apply radical rule:

a b = a \cdot b

513 = 5 \cdot 13

= 3 5 \cdot 13

Multiply the numbers:

5 \cdot 13 = 65

= 365

21313 = 26

21313

Apply radical rule:

a a = a

1313 = 13

= 2 \cdot 13

Multiply the numbers:

2 \cdot 13 = 26

= 26

= - \frac{3 65}{26}

= - \frac{3 65}{26}

u = \frac{3 65}{26}, u = - \frac{3 65}{26}

Substitute back

u = sin (x)

sin (x) = \frac{3 65}{26}, sin (x) = - \frac{3 65}{26}

sin (x) = \frac{3 65}{26}, sin (x) = - \frac{3 65}{26}

sin (x) = \frac{3 65}{26} : x = arcsin (\frac{3 65}{26}) + 2 πn, x = π - arcsin (\frac{3 65}{26}) + 2 πn

sin (x) = \frac{3 65}{26}

Apply trig inverse properties

sin (x) = \frac{3 65}{26}

General solutions for

sin (x) = \frac{3 65}{26}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

x = arcsin (\frac{3 65}{26}) + 2 πn, x = π - arcsin (\frac{3 65}{26}) + 2 πn

x = arcsin (\frac{3 65}{26}) + 2 πn, x = π - arcsin (\frac{3 65}{26}) + 2 πn

sin (x) = - \frac{3 65}{26} : x = arcsin (- \frac{3 65}{26}) + 2 πn, x = π + arcsin (\frac{3 65}{26}) + 2 πn

sin (x) = - \frac{3 65}{26}

Apply trig inverse properties

sin (x) = - \frac{3 65}{26}

General solutions for

sin (x) = - \frac{3 65}{26}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

x = arcsin (- \frac{3 65}{26}) + 2 πn, x = π + arcsin (\frac{3 65}{26}) + 2 πn

x = arcsin (- \frac{3 65}{26}) + 2 πn, x = π + arcsin (\frac{3 65}{26}) + 2 πn

Combine all the solutions

x = arcsin (\frac{3 65}{26}) + 2 πn, x = π - arcsin (\frac{3 65}{26}) + 2 πn, x = arcsin (- \frac{3 65}{26}) + 2 πn, x = π + arcsin (\frac{3 65}{26}) + 2 πn

Show solutions in decimal form

x = 1.19512 \dots + 2 πn, x = π - 1.19512 \dots + 2 πn, x = - 1.19512 \dots + 2 πn, x = π + 1.19512 \dots + 2 πn

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