What is the general solution for sin(θ)+cos(θ)= 7/5 ,sin(2θ) ?

The general solution for sin(θ)+cos(θ)= 7/5 ,sin(2θ) is No Solution for θ\in\mathbb{R}

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sin(θ)+cos(θ)= 7/5 ,sin(2θ)

Solution

sin (θ) + cos (θ) = \frac{7}{5}, sin (2 θ)

Solution

No Solution for θ \in R

Solution steps

sin (θ) + cos (θ) = \frac{7}{5}, sin (2 θ)

Rewrite using trig identities

sin (θ) + cos (θ)

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

sin (θ) + cos (θ)

Rewrite as

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

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 (θ) + sin (\frac{π}{4}) cos (θ))

Use the Angle Sum identity:

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

= 2 sin (θ + \frac{π}{4})

= 2 sin (θ + \frac{π}{4})

2 sin (θ + \frac{π}{4}) = \frac{7}{5}

Divide both sides by

2

2 sin (θ + \frac{π}{4}) = \frac{7}{5}

Divide both sides by

2

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{\frac{7}{5}}{2}

Simplify

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{\frac{7}{5}}{2}

Simplify

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

\frac{2 sin ( θ + \frac{π}{4} )}{2}

Cancel the common factor:

2

= sin (θ + \frac{π}{4})

Simplify

\frac{\frac{7}{5}}{2} : \frac{7 2}{10}

\frac{\frac{7}{5}}{2}

Apply the fraction rule:

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

= \frac{7}{5 2}

Rationalize

\frac{7}{5 2} : \frac{7 2}{10}

\frac{7}{5 2}

Multiply by the conjugate

\frac{2}{2}

= \frac{7 2}{5 2 2}

522 = 10

522

Apply radical rule:

a a = a

22 = 2

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= \frac{7 2}{10}

= \frac{7 2}{10}

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

Apply trig inverse properties

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

General solutions for

sin (θ + \frac{π}{4}) = \frac{7 2}{10}

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

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn, θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn, θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn

Solve

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn : θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = arcsin (\frac{7 2}{10}) + 2 πn

Simplify

arcsin (\frac{7 2}{10}) + 2 πn : arcsin (\frac{7}{5 2}) + 2 πn

arcsin (\frac{7 2}{10}) + 2 πn

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

Cancel

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{7}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{7}{5 2}

= \frac{7}{2 \cdot 5}

= arcsin (\frac{7}{5 2}) + 2 πn

θ + \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn

Move

\frac{π}{4}

to the right side

θ + \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn

Subtract

\frac{π}{4}

from both sides

θ + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Simplify

θ + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Simplify

θ + \frac{π}{4} - \frac{π}{4} : θ

θ + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= θ

Simplify

arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4} : arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

= arcsin (\frac{7 2}{10}) + 2 πn - \frac{π}{4}

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

Cancel

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{7}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{7}{5 2}

= \frac{7}{2 \cdot 5}

= arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Could not simplify further

= arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Solve

θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn : θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = π - arcsin (\frac{7 2}{10}) + 2 πn

Simplify

π - arcsin (\frac{7 2}{10}) + 2 πn : π - arcsin (\frac{7}{5 2}) + 2 πn

π - arcsin (\frac{7 2}{10}) + 2 πn

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

Cancel

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{7}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{7}{5 2}

= \frac{7}{2 \cdot 5}

= π - arcsin (\frac{7}{5 2}) + 2 πn

θ + \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn

Move

\frac{π}{4}

to the right side

θ + \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn

Subtract

\frac{π}{4}

from both sides

θ + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Simplify

θ + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Simplify

θ + \frac{π}{4} - \frac{π}{4} : θ

θ + \frac{π}{4} - \frac{π}{4}

Add similar elements:

\frac{π}{4} - \frac{π}{4} = 0

= θ

Simplify

π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4} : π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

= π - arcsin (\frac{7 2}{10}) + 2 πn - \frac{π}{4}

\frac{7 2}{10} = \frac{7}{5 2}

\frac{7 2}{10}

Factor

10 : 2 \cdot 5

Factor

10 = 2 \cdot 5

= \frac{7 2}{2 \cdot 5}

Cancel

\frac{7 2}{2 \cdot 5} : \frac{7}{2 \cdot 5}

\frac{7 2}{2 \cdot 5}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{7 \cdot 2 ^{\frac{1}{2}}}{2 \cdot 5}

Apply exponent rule:

\frac{x ^{a}}{x ^{b}} = \frac{1}{x ^{b - a}}

\frac{2 ^{\frac{1}{2}}}{2 ^{1}} = \frac{1}{2 ^{1 - \frac{1}{2}}}

= \frac{7}{5 \cdot 2 ^{- \frac{1}{2} + 1}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= \frac{7}{5 \cdot 2 ^{\frac{1}{2}}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{7}{5 2}

= \frac{7}{2 \cdot 5}

= π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Could not simplify further

= π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}, θ = π - arcsin (\frac{7}{5 2}) + 2 πn - \frac{π}{4}

Solutions for the range

sin (2 θ)

No Solution for θ \in R

Popular Examples

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)+cos(θ)= 7/5 ,sin(2θ) ?
The general solution for sin(θ)+cos(θ)= 7/5 ,sin(2θ) is No Solution for θ\in\mathbb{R}