What is the general solution for sin^2(x)=((10m-7))/9 ?

The general solution for sin^2(x)=((10m-7))/9 is

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Popular Trigonometry >

sin^2(x)=((10m-7))/9

Solution

sin^{2} (x) = \frac{( 10 m - 7 )}{9}

Solution

x = arcsin (\frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (- \frac{10 m - 7}{3}) + 2 πn, x = arcsin (- \frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (\frac{10 m - 7}{3}) + 2 πn

Solution steps

sin^{2} (x) = \frac{( 10 m - 7 )}{9}

Solve by substitution

sin^{2} (x) = \frac{10 m - 7}{9}

Let:

sin (x) = u

u^{2} = \frac{10 m - 7}{9}

u^{2} = \frac{10 m - 7}{9} : u = \frac{10 m - 7}{3}, u = - \frac{10 m - 7}{3}

u^{2} = \frac{10 m - 7}{9}

For

x^{2} = f (a)

the solutions are

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

u = \frac{10 m - 7}{9}, u = - \frac{10 m - 7}{9}

Simplify

\frac{10 m - 7}{9} : \frac{10 m - 7}{3}

\frac{10 m - 7}{9}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{10 m - 7}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{10 m - 7}{3}

Simplify

- \frac{10 m - 7}{9} : - \frac{10 m - 7}{3}

- \frac{10 m - 7}{9}

Simplify

\frac{10 m - 7}{9} : \frac{10 m - 7}{3}

\frac{10 m - 7}{9}

Apply radical rule:

n \frac{a}{b} = \frac{n a}{n b},

assuming

a \geq 0, b \geq 0

= \frac{10 m - 7}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

n a^{n} = a

3^{2} = 3

= 3

= \frac{10 m - 7}{3}

= - \frac{10 m - 7}{3}

u = \frac{10 m - 7}{3}, u = - \frac{10 m - 7}{3}

Substitute back

u = sin (x)

sin (x) = \frac{10 m - 7}{3}, sin (x) = - \frac{10 m - 7}{3}

sin (x) = \frac{10 m - 7}{3}, sin (x) = - \frac{10 m - 7}{3}

sin (x) = \frac{10 m - 7}{3} : x = arcsin (\frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (- \frac{10 m - 7}{3}) + 2 πn

sin (x) = \frac{10 m - 7}{3}

Apply trig inverse properties

sin (x) = \frac{10 m - 7}{3}

General solutions for

sin (x) = \frac{10 m - 7}{3}

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

x = arcsin (\frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (- \frac{10 m - 7}{3}) + 2 πn

x = arcsin (\frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (- \frac{10 m - 7}{3}) + 2 πn

sin (x) = - \frac{10 m - 7}{3} : x = arcsin (- \frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (\frac{10 m - 7}{3}) + 2 πn

sin (x) = - \frac{10 m - 7}{3}

Apply trig inverse properties

sin (x) = - \frac{10 m - 7}{3}

General solutions for

sin (x) = - \frac{10 m - 7}{3}

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

x = arcsin (- \frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (\frac{10 m - 7}{3}) + 2 πn

x = arcsin (- \frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (\frac{10 m - 7}{3}) + 2 πn

Combine all the solutions

x = arcsin (\frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (- \frac{10 m - 7}{3}) + 2 πn, x = arcsin (- \frac{10 m - 7}{3}) + 2 πn, x = π + arcsin (\frac{10 m - 7}{3}) + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sin^2(x)=((10m-7))/9 ?
The general solution for sin^2(x)=((10m-7))/9 is