What is the general solution for cos(a)=sin(a+30) ?

The general solution for cos(a)=sin(a+30) is a=30+180n

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cos(a)=sin(a+30)

Solution

cos (a) = sin (a + 3 0^{\circ})

Solution

a = 3 0^{\circ} + 18 0^{\circ} n

Radians

a = \frac{π}{6} + πn

Solution steps

cos (a) = sin (a + 3 0^{\circ})

Rewrite using trig identities

cos (a) = sin (a + 3 0^{\circ})

Rewrite using trig identities

sin (a + 3 0^{\circ})

Use the Angle Sum identity:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (a) cos (3 0^{\circ}) + cos (a) sin (3 0^{\circ})

Simplify

sin (a) cos (3 0^{\circ}) + cos (a) sin (3 0^{\circ}) : \frac{3}{2} sin (a) + \frac{1}{2} cos (a)

sin (a) cos (3 0^{\circ}) + cos (a) sin (3 0^{\circ})

Simplify

cos (3 0^{\circ}) : \frac{3}{2}

cos (3 0^{\circ})

Use the following trivial identity:

cos (3 0^{\circ}) = \frac{3}{2}

cos (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{3}{2}

= \frac{3}{2} sin (a) + sin (3 0^{\circ}) cos (a)

Simplify

sin (3 0^{\circ}) : \frac{1}{2}

sin (3 0^{\circ})

Use the following trivial identity:

sin (3 0^{\circ}) = \frac{1}{2}

sin (x)

periodicity table with

36 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

= \frac{3}{2} sin (a) + \frac{1}{2} cos (a)

= \frac{3}{2} sin (a) + \frac{1}{2} cos (a)

cos (a) = \frac{3}{2} sin (a) + \frac{1}{2} cos (a)

cos (a) = \frac{3}{2} sin (a) + \frac{1}{2} cos (a)

Subtract

\frac{3}{2} sin (a) + \frac{1}{2} cos (a)

from both sides

\frac{1}{2} cos (a) - \frac{3}{2} sin (a) = 0

Simplify

\frac{1}{2} cos (a) - \frac{3}{2} sin (a) : \frac{cos ( a ) - 3 sin ( a )}{2}

\frac{1}{2} cos (a) - \frac{3}{2} sin (a)

\frac{1}{2} cos (a) = \frac{cos ( a )}{2}

\frac{1}{2} cos (a)

Multiply fractions:

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

= \frac{1 \cdot cos ( a )}{2}

Multiply:

1 \cdot cos (a) = cos (a)

= \frac{cos ( a )}{2}

\frac{3}{2} sin (a) = \frac{3 sin ( a )}{2}

\frac{3}{2} sin (a)

Multiply fractions:

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

= \frac{3 sin ( a )}{2}

= \frac{cos ( a )}{2} - \frac{3 sin ( a )}{2}

Apply rule

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

= \frac{cos ( a ) - 3 sin ( a )}{2}

\frac{cos ( a ) - 3 sin ( a )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

cos (a) - 3 sin (a) = 0

Rewrite using trig identities

cos (a) - 3 sin (a) = 0

Divide both sides by

cos (a), cos (a) \neq = 0

\frac{cos ( a ) - 3 sin ( a )}{cos ( a )} = \frac{0}{cos ( a )}

Simplify

1 - \frac{3 sin ( a )}{cos ( a )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 - 3 tan (a) = 0

1 - 3 tan (a) = 0

Move

1

to the right side

1 - 3 tan (a) = 0

Subtract

1

from both sides

1 - 3 tan (a) - 1 = 0 - 1

Simplify

- 3 tan (a) = - 1

- 3 tan (a) = - 1

Divide both sides by

- 3

- 3 tan (a) = - 1

Divide both sides by

- 3

\frac{- 3 tan ( a )}{- 3} = \frac{- 1}{- 3}

Simplify

\frac{- 3 tan ( a )}{- 3} = \frac{- 1}{- 3}

Simplify

\frac{- 3 tan ( a )}{- 3} : tan (a)

\frac{- 3 tan ( a )}{- 3}

Apply the fraction rule:

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

= \frac{3 tan ( a )}{3}

Cancel the common factor:

3

= tan (a)

Simplify

\frac{- 1}{- 3} : \frac{3}{3}

\frac{- 1}{- 3}

Apply the fraction rule:

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

= \frac{1}{3}

Rationalize

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

= \frac{1 \cdot 3}{3 3}

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

= \frac{3}{3}

tan (a) = \frac{3}{3}

tan (a) = \frac{3}{3}

tan (a) = \frac{3}{3}

General solutions for

tan (a) = \frac{3}{3}

tan (x)

periodicity table with

18 0^{\circ} n

cycle:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

a = 3 0^{\circ} + 18 0^{\circ} n

a = 3 0^{\circ} + 18 0^{\circ} n

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Frequently Asked Questions (FAQ)

What is the general solution for cos(a)=sin(a+30) ?
The general solution for cos(a)=sin(a+30) is a=30+180n