What is the general solution for sin(θ)+cos(θ)=0.46 ?

The general solution for sin(θ)+cos(θ)=0.46 is θ=0.33129…+2pin-pi/4 ,θ=pi-0.33129…+2pin-pi/4

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sin(θ)+cos(θ)=0.46

Solution

sin (θ) + cos (θ) = 0.46

Solution

θ = 0.33129 \dots + 2 πn - \frac{π}{4}, θ = π - 0.33129 \dots + 2 πn - \frac{π}{4}

Degrees

θ = - 26.01811 \dots^{\circ} + 36 0^{\circ} n, θ = 116.01811 \dots^{\circ} + 36 0^{\circ} n

Solution steps

sin (θ) + cos (θ) = 0.46

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}) = 0.46

Divide both sides by

2

2 sin (θ + \frac{π}{4}) = 0.46

Divide both sides by

2

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{0.46}{2}

Simplify

\frac{2 sin ( θ + \frac{π}{4} )}{2} = \frac{0.46}{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{0.46}{2} : 0.32526 \dots

\frac{0.46}{2}

Convert element to a decimal form

2 = 1.41421 \dots

= \frac{0.46}{1.41421 \dots}

Divide the numbers:

\frac{0.46}{1.41421 \dots} = 0.32526 \dots

= 0.32526 \dots

sin (θ + \frac{π}{4}) = 0.32526 \dots

sin (θ + \frac{π}{4}) = 0.32526 \dots

sin (θ + \frac{π}{4}) = 0.32526 \dots

Apply trig inverse properties

sin (θ + \frac{π}{4}) = 0.32526 \dots

General solutions for

sin (θ + \frac{π}{4}) = 0.32526 \dots

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

θ + \frac{π}{4} = arcsin (0.32526 \dots) + 2 πn, θ + \frac{π}{4} = π - arcsin (0.32526 \dots) + 2 πn

θ + \frac{π}{4} = arcsin (0.32526 \dots) + 2 πn, θ + \frac{π}{4} = π - arcsin (0.32526 \dots) + 2 πn

Solve

θ + \frac{π}{4} = arcsin (0.32526 \dots) + 2 πn : θ = arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = arcsin (0.32526 \dots) + 2 πn

Move

\frac{π}{4}

to the right side

θ + \frac{π}{4} = arcsin (0.32526 \dots) + 2 πn

Subtract

\frac{π}{4}

from both sides

θ + \frac{π}{4} - \frac{π}{4} = arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

θ = arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

θ = arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Solve

θ + \frac{π}{4} = π - arcsin (0.32526 \dots) + 2 πn : θ = π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

θ + \frac{π}{4} = π - arcsin (0.32526 \dots) + 2 πn

Move

\frac{π}{4}

to the right side

θ + \frac{π}{4} = π - arcsin (0.32526 \dots) + 2 πn

Subtract

\frac{π}{4}

from both sides

θ + \frac{π}{4} - \frac{π}{4} = π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Simplify

θ = π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

θ = π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

θ = arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}, θ = π - arcsin (0.32526 \dots) + 2 πn - \frac{π}{4}

Show solutions in decimal form

θ = 0.33129 \dots + 2 πn - \frac{π}{4}, θ = π - 0.33129 \dots + 2 πn - \frac{π}{4}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(θ)+cos(θ)=0.46 ?
The general solution for sin(θ)+cos(θ)=0.46 is θ=0.33129…+2pin-pi/4 ,θ=pi-0.33129…+2pin-pi/4