What is the general solution for sin(b)=cos(2b-30) ?

The general solution for sin(b)=cos(2b-30) is b=(pi+60+4pin)/6 ,b=-(pi-60+4pin)/2

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sin(b)=cos(2b-30)

Solution

sin (b) = cos (2 b - 30)

Solution

b = \frac{π + 60 + 4 πn}{6}, b = - \frac{π - 60 + 4 πn}{2}

Degrees

b = 602.95779 \dots^{\circ} + 12 0^{\circ} n, b = 1628.87338 \dots^{\circ} - 36 0^{\circ} n

Solution steps

sin (b) = cos (2 b - 30)

Rewrite using trig identities

sin (b) = cos (2 b - 30)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

sin (b) = sin (\frac{π}{2} - (2 b - 30))

sin (b) = sin (\frac{π}{2} - (2 b - 30))

Apply trig inverse properties

sin (b) = sin (\frac{π}{2} - (2 b - 30))

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

b = \frac{π}{2} - (2 b - 30) + 2 πn, b = π - (\frac{π}{2} - (2 b - 30)) + 2 πn

b = \frac{π}{2} - (2 b - 30) + 2 πn, b = π - (\frac{π}{2} - (2 b - 30)) + 2 πn

b = \frac{π}{2} - (2 b - 30) + 2 πn : b = \frac{π + 60 + 4 πn}{6}

b = \frac{π}{2} - (2 b - 30) + 2 πn

Expand

\frac{π}{2} - (2 b - 30) + 2 πn : \frac{π}{2} - 2 b + 30 + 2 πn

\frac{π}{2} - (2 b - 30) + 2 πn

- (2 b - 30) : - 2 b + 30

- (2 b - 30)

Distribute parentheses

= - (2 b) - (- 30)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 b + 30

= \frac{π}{2} - 2 b + 30 + 2 πn

b = \frac{π}{2} - 2 b + 30 + 2 πn

Move

2 b

to the left side

b = \frac{π}{2} - 2 b + 30 + 2 πn

Add

2 b

to both sides

b + 2 b = \frac{π}{2} - 2 b + 30 + 2 πn + 2 b

Simplify

3 b = \frac{π}{2} + 30 + 2 πn

3 b = \frac{π}{2} + 30 + 2 πn

Divide both sides by

3

3 b = \frac{π}{2} + 30 + 2 πn

Divide both sides by

3

\frac{3 b}{3} = \frac{\frac{π}{2}}{3} + \frac{30}{3} + \frac{2 πn}{3}

Simplify

\frac{3 b}{3} = \frac{\frac{π}{2}}{3} + \frac{30}{3} + \frac{2 πn}{3}

Simplify

\frac{3 b}{3} : b

\frac{3 b}{3}

Divide the numbers:

\frac{3}{3} = 1

= b

Simplify

\frac{\frac{π}{2}}{3} + \frac{30}{3} + \frac{2 πn}{3} : \frac{π + 60 + 4 πn}{6}

\frac{\frac{π}{2}}{3} + \frac{30}{3} + \frac{2 πn}{3}

Apply rule

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

= \frac{\frac{π}{2} + 30 + 2 πn}{3}

Join

\frac{π}{2} + 30 + 2 πn : \frac{π + 60 + 4 πn}{2}

\frac{π}{2} + 30 + 2 πn

Convert element to fraction:

30 = \frac{30 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π}{2} + \frac{30 \cdot 2}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π + 30 \cdot 2 + 2 πn \cdot 2}{2}

π + 30 \cdot 2 + 2 πn \cdot 2 = π + 60 + 4 πn

π + 30 \cdot 2 + 2 πn \cdot 2

Multiply the numbers:

30 \cdot 2 = 60

= π + 60 + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π + 60 + 4 πn

= \frac{π + 60 + 4 πn}{2}

= \frac{\frac{π + 60 + 4 πn}{2}}{3}

Apply the fraction rule:

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

= \frac{π + 60 + 4 πn}{2 \cdot 3}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{π + 60 + 4 πn}{6}

b = \frac{π + 60 + 4 πn}{6}

b = \frac{π + 60 + 4 πn}{6}

b = \frac{π + 60 + 4 πn}{6}

b = π - (\frac{π}{2} - (2 b - 30)) + 2 πn : b = - \frac{π - 60 + 4 πn}{2}

b = π - (\frac{π}{2} - (2 b - 30)) + 2 πn

Expand

π - (\frac{π}{2} - (2 b - 30)) + 2 πn : π - \frac{π}{2} + 2 b - 30 + 2 πn

π - (\frac{π}{2} - (2 b - 30)) + 2 πn

- (2 b - 30) : - 2 b + 30

- (2 b - 30)

Distribute parentheses

= - (2 b) - (- 30)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 2 b + 30

= π - (- 2 b + 30 + \frac{π}{2}) + 2 πn

- (\frac{π}{2} - 2 b + 30) : - \frac{π}{2} + 2 b - 30

- (\frac{π}{2} - 2 b + 30)

Distribute parentheses

= - (\frac{π}{2}) - (- 2 b) - (30)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + 2 b - 30

= π - \frac{π}{2} + 2 b - 30 + 2 πn

b = π - \frac{π}{2} + 2 b - 30 + 2 πn

Move

2 b

to the left side

b = π - \frac{π}{2} + 2 b - 30 + 2 πn

Subtract

2 b

from both sides

b - 2 b = π - \frac{π}{2} + 2 b - 30 + 2 πn - 2 b

Simplify

- b = π - \frac{π}{2} - 30 + 2 πn

- b = π - \frac{π}{2} - 30 + 2 πn

Divide both sides by

- 1

- b = π - \frac{π}{2} - 30 + 2 πn

Divide both sides by

- 1

\frac{- b}{- 1} = \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} - \frac{30}{- 1} + \frac{2 πn}{- 1}

Simplify

\frac{- b}{- 1} = \frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} - \frac{30}{- 1} + \frac{2 πn}{- 1}

Simplify

\frac{- b}{- 1} : b

\frac{- b}{- 1}

Apply the fraction rule:

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

= \frac{b}{1}

Apply rule

\frac{a}{1} = a

= b

Simplify

\frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} - \frac{30}{- 1} + \frac{2 πn}{- 1} : - \frac{π - 60 + 4 πn}{2}

\frac{π}{- 1} - \frac{\frac{π}{2}}{- 1} - \frac{30}{- 1} + \frac{2 πn}{- 1}

Apply rule

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

= \frac{π - \frac{π}{2} - 30 + 2 πn}{- 1}

Apply the fraction rule:

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

= - \frac{π - \frac{π}{2} - 30 + 2 πn}{1}

Join

π - \frac{π}{2} - 30 + 2 πn : \frac{π - 60 + 4 πn}{2}

π - \frac{π}{2} - 30 + 2 πn

Convert element to fraction:

π = \frac{π 2}{2}, 30 = \frac{30 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 2}{2} - \frac{π}{2} - \frac{30 \cdot 2}{2} + \frac{2 πn \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 - π - 30 \cdot 2 + 2 πn \cdot 2}{2}

π 2 - π - 30 \cdot 2 + 2 πn \cdot 2 = π - 60 + 4 πn

π 2 - π - 30 \cdot 2 + 2 πn \cdot 2

Add similar elements:

2 π - π = π

= π - 30 \cdot 2 + 2 \cdot 2 πn

Multiply the numbers:

30 \cdot 2 = 60

= π - 60 + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= π - 60 + 4 πn

= \frac{π - 60 + 4 πn}{2}

= - \frac{\frac{π + 4 πn - 60}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= - \frac{π + 4 πn - 60}{2}

b = - \frac{π - 60 + 4 πn}{2}

b = - \frac{π - 60 + 4 πn}{2}

b = - \frac{π - 60 + 4 πn}{2}

b = \frac{π + 60 + 4 πn}{6}, b = - \frac{π - 60 + 4 πn}{2}

b = \frac{π + 60 + 4 πn}{6}, b = - \frac{π - 60 + 4 πn}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(b)=cos(2b-30) ?
The general solution for sin(b)=cos(2b-30) is b=(pi+60+4pin)/6 ,b=-(pi-60+4pin)/2