What is the general solution for (sin(a)+cos(a))^2=1+2,acos(a) ?

The general solution for (sin(a)+cos(a))^2=1+2,acos(a) is No Solution for a\in\mathbb{R}

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Popular >

(sin(a)+cos(a))^2=1+2,acos(a)

Solution

(sin (a) + cos (a))^{2} = 1 + 2, a cos (a)

Solution

No Solution for a \in R

Solution steps

(sin (a) + cos (a))^{2} = 1 + 2, a cos (a)

Subtract

1 + 2

from both sides

(sin (a) + cos (a))^{2} - 3 = 0

Factor

(sin (a) + cos (a))^{2} - 3 : (sin (a) + cos (a) + 3) (sin (a) + cos (a) - 3)

(sin (a) + cos (a))^{2} - 3

Apply radical rule:

a = (a)^{2}

3 = (3)^{2}

= (sin (a) + cos (a))^{2} - (3)^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

(sin (a) + cos (a))^{2} - (3)^{2} = ((sin (a) + cos (a)) + 3) ((sin (a) + cos (a)) - 3)

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

Refine

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

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

Solving each part separately

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

sin (a) + cos (a) + 3 = 0, a cos (a) :

No Solution

sin (a) + cos (a) + 3 = 0, a cos (a)

Rewrite using trig identities

sin (a) + cos (a) + 3

sin (a) + cos (a) = 2 sin (a + \frac{π}{4})

sin (a) + cos (a)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (a) + sin (\frac{π}{4}) cos (a))

Use the Angle Sum identity:

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

= 2 sin (a + \frac{π}{4})

= 3 + 2 sin (a + \frac{π}{4})

3 + 2 sin (a + \frac{π}{4}) = 0

Move

3

to the right side

3 + 2 sin (a + \frac{π}{4}) = 0

Subtract

3

from both sides

3 + 2 sin (a + \frac{π}{4}) - 3 = 0 - 3

Simplify

2 sin (a + \frac{π}{4}) = - 3

2 sin (a + \frac{π}{4}) = - 3

Divide both sides by

2

2 sin (a + \frac{π}{4}) = - 3

Divide both sides by

2

\frac{2 sin ( a + \frac{π}{4} )}{2} = \frac{- 3}{2}

Simplify

\frac{2 sin ( a + \frac{π}{4} )}{2} = \frac{- 3}{2}

Simplify

\frac{2 sin ( a + \frac{π}{4} )}{2} : sin (a + \frac{π}{4})

\frac{2 sin ( a + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (a + \frac{π}{4})

Simplify

\frac{- 3}{2} : - \frac{3}{2}

\frac{- 3}{2}

Apply the fraction rule:

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

= - \frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= - \frac{3}{2}

sin (a + \frac{π}{4}) = - \frac{3}{2}

sin (a + \frac{π}{4}) = - \frac{3}{2}

sin (a + \frac{π}{4}) = - \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

sin (a) + cos (a) - 3 = 0, a cos (a) :

No Solution

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

Rewrite using trig identities

sin (a) + cos (a) - 3

sin (a) + cos (a) = 2 sin (a + \frac{π}{4})

sin (a) + cos (a)

Rewrite as

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

Use the following trivial identity:

cos (\frac{π}{4}) = \frac{1}{2}

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{1}{2}

= 2 (cos (\frac{π}{4}) sin (a) + sin (\frac{π}{4}) cos (a))

Use the Angle Sum identity:

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

= 2 sin (a + \frac{π}{4})

= - 3 + 2 sin (a + \frac{π}{4})

- 3 + 2 sin (a + \frac{π}{4}) = 0

Move

3

to the right side

- 3 + 2 sin (a + \frac{π}{4}) = 0

Add

3

to both sides

- 3 + 2 sin (a + \frac{π}{4}) + 3 = 0 + 3

Simplify

2 sin (a + \frac{π}{4}) = 3

2 sin (a + \frac{π}{4}) = 3

Divide both sides by

2

2 sin (a + \frac{π}{4}) = 3

Divide both sides by

2

\frac{2 sin ( a + \frac{π}{4} )}{2} = \frac{3}{2}

Simplify

\frac{2 sin ( a + \frac{π}{4} )}{2} = \frac{3}{2}

Simplify

\frac{2 sin ( a + \frac{π}{4} )}{2} : sin (a + \frac{π}{4})

\frac{2 sin ( a + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (a + \frac{π}{4})

Simplify

\frac{3}{2} : \frac{3}{2}

\frac{3}{2}

Combine same powers :

\frac{x}{y} = \frac{x}{y}

= \frac{3}{2}

sin (a + \frac{π}{4}) = \frac{3}{2}

sin (a + \frac{π}{4}) = \frac{3}{2}

sin (a + \frac{π}{4}) = \frac{3}{2}

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

No Solution for a \in R

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Frequently Asked Questions (FAQ)

What is the general solution for (sin(a)+cos(a))^2=1+2,acos(a) ?
The general solution for (sin(a)+cos(a))^2=1+2,acos(a) is No Solution for a\in\mathbb{R}