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Popular >

sin^2(a-pi/8)= 2/3

Solution

sin^{2} (a - \frac{π}{8}) = \frac{2}{3}

Solution

a = 0.95531 \dots + 2 πn + \frac{π}{8}, a = π - 0.95531 \dots + 2 πn + \frac{π}{8}, a = - 0.95531 \dots + 2 πn + \frac{π}{8}, a = π + 0.95531 \dots + 2 πn + \frac{π}{8}

Degrees

a = 77.23561 \dots^{\circ} + 36 0^{\circ} n, a = 147.76438 \dots^{\circ} + 36 0^{\circ} n, a = - 32.23561 \dots^{\circ} + 36 0^{\circ} n, a = 257.23561 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin^{2} (a - \frac{π}{8}) = \frac{2}{3}

Solve by substitution

sin^{2} (a - \frac{π}{8}) = \frac{2}{3}

Let:

sin (a - \frac{π}{8}) = u

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

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

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

For

x^{2} = f (a)

the solutions are

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

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

Substitute back

u = sin (a - \frac{π}{8})

sin (a - \frac{π}{8}) = \frac{2}{3}, sin (a - \frac{π}{8}) = - \frac{2}{3}

sin (a - \frac{π}{8}) = \frac{2}{3}, sin (a - \frac{π}{8}) = - \frac{2}{3}

sin (a - \frac{π}{8}) = \frac{2}{3} : a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

sin (a - \frac{π}{8}) = \frac{2}{3}

Apply trig inverse properties

sin (a - \frac{π}{8}) = \frac{2}{3}

General solutions for

sin (a - \frac{π}{8}) = \frac{2}{3}

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

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn, a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn, a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

Solve

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn : a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn

Move

\frac{π}{8}

to the right side

a - \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn

Add

\frac{π}{8}

to both sides

a - \frac{π}{8} + \frac{π}{8} = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Simplify

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Solve

a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn : a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

Move

\frac{π}{8}

to the right side

a - \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn

Add

\frac{π}{8}

to both sides

a - \frac{π}{8} + \frac{π}{8} = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Simplify

a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

sin (a - \frac{π}{8}) = - \frac{2}{3} : a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

sin (a - \frac{π}{8}) = - \frac{2}{3}

Apply trig inverse properties

sin (a - \frac{π}{8}) = - \frac{2}{3}

General solutions for

sin (a - \frac{π}{8}) = - \frac{2}{3}

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

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn, a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn, a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

Solve

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn : a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = arcsin (- \frac{2}{3}) + 2 πn

Simplify

arcsin (- \frac{2}{3}) + 2 πn : - arcsin (\frac{2}{3}) + 2 πn

arcsin (- \frac{2}{3}) + 2 πn

Use the following property:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{2}{3}) = - arcsin (\frac{2}{3})

= - arcsin (\frac{2}{3}) + 2 πn

a - \frac{π}{8} = - arcsin (\frac{2}{3}) + 2 πn

Move

\frac{π}{8}

to the right side

a - \frac{π}{8} = - arcsin (\frac{2}{3}) + 2 πn

Add

\frac{π}{8}

to both sides

a - \frac{π}{8} + \frac{π}{8} = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Simplify

a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Solve

a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn : a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

Move

\frac{π}{8}

to the right side

a - \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn

Add

\frac{π}{8}

to both sides

a - \frac{π}{8} + \frac{π}{8} = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Simplify

a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Combine all the solutions

a = arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = - arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}, a = π + arcsin (\frac{2}{3}) + 2 πn + \frac{π}{8}

Show solutions in decimal form

a = 0.95531 \dots + 2 πn + \frac{π}{8}, a = π - 0.95531 \dots + 2 πn + \frac{π}{8}, a = - 0.95531 \dots + 2 πn + \frac{π}{8}, a = π + 0.95531 \dots + 2 πn + \frac{π}{8}

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