What is the general solution for cos(7a)=sin(a-6) ?

The general solution for cos(7a)=sin(a-6) is a=(12+4pin+pi)/(16),a=-(pi+4pin+12)/(12)

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cos(7a)=sin(a-6)

Solution

cos (7 a) = sin (a - 6)

Solution

a = \frac{12 + 4 πn + π}{16}, a = - \frac{π + 4 πn + 12}{12}

Degrees

a = 54.22183 \dots^{\circ} + 4 5^{\circ} n, a = - 72.29577 \dots^{\circ} - 6 0^{\circ} n

Solution steps

cos (7 a) = sin (a - 6)

Rewrite using trig identities

cos (7 a) = sin (a - 6)

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

cos (7 a) = sin (\frac{π}{2} - 7 a)

cos (7 a) = sin (\frac{π}{2} - 7 a)

Apply trig inverse properties

cos (7 a) = sin (\frac{π}{2} - 7 a)

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

a - 6 = \frac{π}{2} - 7 a + 2 πn, a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn

a - 6 = \frac{π}{2} - 7 a + 2 πn, a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn

a - 6 = \frac{π}{2} - 7 a + 2 πn : a = \frac{12 + 4 πn + π}{16}

a - 6 = \frac{π}{2} - 7 a + 2 πn

Move

6

to the right side

a - 6 = \frac{π}{2} - 7 a + 2 πn

Add

6

to both sides

a - 6 + 6 = \frac{π}{2} - 7 a + 2 πn + 6

Simplify

a = \frac{π}{2} - 7 a + 2 πn + 6

a = \frac{π}{2} - 7 a + 2 πn + 6

Move

7 a

to the left side

a = \frac{π}{2} - 7 a + 2 πn + 6

Add

7 a

to both sides

a + 7 a = \frac{π}{2} - 7 a + 2 πn + 6 + 7 a

Simplify

8 a = \frac{π}{2} + 2 πn + 6

8 a = \frac{π}{2} + 2 πn + 6

Divide both sides by

8

8 a = \frac{π}{2} + 2 πn + 6

Divide both sides by

8

\frac{8 a}{8} = \frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8}

Simplify

\frac{8 a}{8} = \frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8}

Simplify

\frac{8 a}{8} : a

\frac{8 a}{8}

Divide the numbers:

\frac{8}{8} = 1

= a

Simplify

\frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8} : \frac{12 + 4 πn + π}{16}

\frac{\frac{π}{2}}{8} + \frac{2 πn}{8} + \frac{6}{8}

Group like terms

= \frac{6}{8} + \frac{2 πn}{8} + \frac{\frac{π}{2}}{8}

Apply rule

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

= \frac{6 + 2 πn + \frac{π}{2}}{8}

Join

6 + 2 πn + \frac{π}{2} : \frac{12 + 4 πn + π}{2}

6 + 2 πn + \frac{π}{2}

Convert element to fraction:

6 = \frac{6 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{6 \cdot 2}{2} + \frac{2 πn \cdot 2}{2} + \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{6 \cdot 2 + 2 πn \cdot 2 + π}{2}

6 \cdot 2 + 2 πn \cdot 2 + π = 12 + 4 πn + π

6 \cdot 2 + 2 πn \cdot 2 + π

Multiply the numbers:

6 \cdot 2 = 12

= 12 + 2 \cdot 2 πn + π

Multiply the numbers:

2 \cdot 2 = 4

= 12 + 4 πn + π

= \frac{12 + 4 πn + π}{2}

= \frac{\frac{12 + 4 πn + π}{2}}{8}

Apply the fraction rule:

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

= \frac{12 + 4 πn + π}{2 \cdot 8}

Multiply the numbers:

2 \cdot 8 = 16

= \frac{12 + 4 πn + π}{16}

a = \frac{12 + 4 πn + π}{16}

a = \frac{12 + 4 πn + π}{16}

a = \frac{12 + 4 πn + π}{16}

a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn : a = - \frac{π + 4 πn + 12}{12}

a - 6 = π - (\frac{π}{2} - 7 a) + 2 πn

Expand

π - (\frac{π}{2} - 7 a) + 2 πn : π - \frac{π}{2} + 7 a + 2 πn

π - (\frac{π}{2} - 7 a) + 2 πn

- (\frac{π}{2} - 7 a) : - \frac{π}{2} + 7 a

- (\frac{π}{2} - 7 a)

Distribute parentheses

= - (\frac{π}{2}) - (- 7 a)

Apply minus-plus rules

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

= - \frac{π}{2} + 7 a

= π - \frac{π}{2} + 7 a + 2 πn

a - 6 = π - \frac{π}{2} + 7 a + 2 πn

Move

6

to the right side

a - 6 = π - \frac{π}{2} + 7 a + 2 πn

Add

6

to both sides

a - 6 + 6 = π - \frac{π}{2} + 7 a + 2 πn + 6

Simplify

a = π - \frac{π}{2} + 7 a + 2 πn + 6

a = π - \frac{π}{2} + 7 a + 2 πn + 6

Move

7 a

to the left side

a = π - \frac{π}{2} + 7 a + 2 πn + 6

Subtract

7 a

from both sides

a - 7 a = π - \frac{π}{2} + 7 a + 2 πn + 6 - 7 a

Simplify

- 6 a = π - \frac{π}{2} + 2 πn + 6

- 6 a = π - \frac{π}{2} + 2 πn + 6

Divide both sides by

- 6

- 6 a = π - \frac{π}{2} + 2 πn + 6

Divide both sides by

- 6

\frac{- 6 a}{- 6} = \frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6}

Simplify

\frac{- 6 a}{- 6} = \frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6}

Simplify

\frac{- 6 a}{- 6} : a

\frac{- 6 a}{- 6}

Apply the fraction rule:

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

= \frac{6 a}{6}

Divide the numbers:

\frac{6}{6} = 1

= a

Simplify

\frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6} : - \frac{π + 4 πn + 12}{12}

\frac{π}{- 6} - \frac{\frac{π}{2}}{- 6} + \frac{2 πn}{- 6} + \frac{6}{- 6}

Group like terms

= \frac{π}{- 6} + \frac{6}{- 6} + \frac{2 πn}{- 6} - \frac{\frac{π}{2}}{- 6}

Apply rule

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

= \frac{π + 6 + 2 πn - \frac{π}{2}}{- 6}

Apply the fraction rule:

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

= - \frac{π + 6 + 2 πn - \frac{π}{2}}{6}

Join

π + 6 + 2 πn - \frac{π}{2} : \frac{π + 4 πn + 12}{2}

π + 6 + 2 πn - \frac{π}{2}

Convert element to fraction:

π = \frac{π 2}{2}, 6 = \frac{6 \cdot 2}{2}, 2 πn = \frac{2 πn 2}{2}

= \frac{π 2}{2} + \frac{6 \cdot 2}{2} + \frac{2 πn \cdot 2}{2} - \frac{π}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π 2 + 6 \cdot 2 + 2 πn \cdot 2 - π}{2}

π 2 + 6 \cdot 2 + 2 πn \cdot 2 - π = π + 4 πn + 12

π 2 + 6 \cdot 2 + 2 πn \cdot 2 - π

Group like terms

= 2 π - π + 2 \cdot 2 πn + 6 \cdot 2

Add similar elements:

2 π - π = π

= π + 2 \cdot 2 πn + 6 \cdot 2

Multiply the numbers:

2 \cdot 2 = 4

= π + 4 πn + 6 \cdot 2

Multiply the numbers:

6 \cdot 2 = 12

= π + 4 πn + 12

= \frac{π + 4 πn + 12}{2}

= - \frac{\frac{π + 4 πn + 12}{2}}{6}

Simplify

\frac{\frac{π + 4 πn + 12}{2}}{6} : \frac{π + 4 πn + 12}{12}

\frac{\frac{π + 4 πn + 12}{2}}{6}

Apply the fraction rule:

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

= \frac{π + 4 πn + 12}{2 \cdot 6}

Multiply the numbers:

2 \cdot 6 = 12

= \frac{π + 4 πn + 12}{12}

= - \frac{π + 4 πn + 12}{12}

a = - \frac{π + 4 πn + 12}{12}

a = - \frac{π + 4 πn + 12}{12}

a = - \frac{π + 4 πn + 12}{12}

a = \frac{12 + 4 πn + π}{16}, a = - \frac{π + 4 πn + 12}{12}

a = \frac{12 + 4 πn + π}{16}, a = - \frac{π + 4 πn + 12}{12}

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Frequently Asked Questions (FAQ)

What is the general solution for cos(7a)=sin(a-6) ?
The general solution for cos(7a)=sin(a-6) is a=(12+4pin+pi)/(16),a=-(pi+4pin+12)/(12)