What is the general solution for sin(θ)+cos(θ)=sqrt(3/5) ?

The general solution for sin(θ)+cos(θ)=sqrt(3/5) is θ=0.57963…+2pin-pi/4 ,θ=pi-0.57963…+2pin-pi/4

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sin(θ)+cos(θ)=sqrt(3/5)

Solution

sin (θ) + cos (θ) = \frac{3}{5}

Solution

θ = 0.57963 \dots + 2 πn - \frac{π}{4}, θ = π - 0.57963 \dots + 2 πn - \frac{π}{4}

Degrees

θ = - 11.78908 \dots^{\circ} + 36 0^{\circ} n, θ = 101.78908 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) + cos (θ) = \frac{3}{5}

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{3}{5}

Divide both sides by

2

2 sin (θ + \frac{π}{4}) = \frac{3}{5}

Divide both sides by

2

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{\frac{3}{5}}{2}

Simplify

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{\frac{3}{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{3}{5}}{2} : \frac{3}{10}

\frac{\frac{3}{5}}{2}

\frac{3}{5} = \frac{3}{5}

\frac{3}{5}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{3}{5}

= \frac{\frac{3}{5}}{2}

Apply the fraction rule:

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

= \frac{3}{5 2}

Simplify

52 : 10

52

Apply radical rule:

a b = a \cdot b

52 = 5 \cdot 2

= 5 \cdot 2

Multiply the numbers:

5 \cdot 2 = 10

= 10

= \frac{3}{10}

Combine same powers :

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

= \frac{3}{10}

sin (θ + \frac{π}{4}) = \frac{3}{10}

sin (θ + \frac{π}{4}) = \frac{3}{10}

sin (θ + \frac{π}{4}) = \frac{3}{10}

Apply trig inverse properties

sin (θ + \frac{π}{4}) = \frac{3}{10}

General solutions for

sin (θ + \frac{π}{4}) = \frac{3}{10}

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

θ + \frac{π}{4} = arcsin (\frac{3}{10}) + 2 πn, θ + \frac{π}{4} = π - arcsin (\frac{3}{10}) + 2 πn

θ + \frac{π}{4} = arcsin (\frac{3}{10}) + 2 πn, θ + \frac{π}{4} = π - arcsin (\frac{3}{10}) + 2 πn

Solve

θ + \frac{π}{4} = arcsin (\frac{3}{10}) + 2 πn : θ = arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = arcsin (\frac{3}{10}) + 2 πn

Move

\frac{π}{4}

to the right side

θ + \frac{π}{4} = arcsin (\frac{3}{10}) + 2 πn

Subtract

\frac{π}{4}

from both sides

θ + \frac{π}{4} - \frac{π}{4} = arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

Simplify

θ = arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

Solve

θ + \frac{π}{4} = π - arcsin (\frac{3}{10}) + 2 πn : θ = π - arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = π - arcsin (\frac{3}{10}) + 2 πn

Move

\frac{π}{4}

to the right side

θ + \frac{π}{4} = π - arcsin (\frac{3}{10}) + 2 πn

Subtract

\frac{π}{4}

from both sides

θ + \frac{π}{4} - \frac{π}{4} = π - arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

Simplify

θ = π - arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

θ = π - arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

θ = arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}, θ = π - arcsin (\frac{3}{10}) + 2 πn - \frac{π}{4}

Show solutions in decimal form

θ = 0.57963 \dots + 2 πn - \frac{π}{4}, θ = π - 0.57963 \dots + 2 πn - \frac{π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)+cos(θ)=sqrt(3/5) ?
The general solution for sin(θ)+cos(θ)=sqrt(3/5) is θ=0.57963…+2pin-pi/4 ,θ=pi-0.57963…+2pin-pi/4