What is the general solution for sin(θ)=0.6367+0.1*cos(θ) ?

The general solution for sin(θ)=0.6367+0.1*cos(θ) is θ=2.55514…+2pin,θ=0.78578…+2pin

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sin(θ)=0.6367+0.1*cos(θ)

sin (θ) = 0.6367 + 0.1 \cdot cos (θ)

θ = 2.55514 \dots + 2 πn, θ = 0.78578 \dots + 2 πn

Solution steps

sin (θ) = 0.6367 + 0.1 cos (θ)

Square both sides

sin^{2} (θ) = (0.6367 + 0.1 cos (θ))^{2}

Subtract

(0.6367 + 0.1 cos (θ))^{2}

from both sides

sin^{2} (θ) - 0.40538689 - 0.12734 cos (θ) - 0.01 cos^{2} (θ) = 0

Rewrite using trig identities

0.59461311 - 0.12734 cos (θ) - 1.01 cos^{2} (θ) = 0

Solve by substitution

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02}, cos (θ) = \frac{2.41845244 - 0.12734}{2.02}

cos (θ) = - \frac{0.12734 + 2.41845244}{2.02} : θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn

cos (θ) = \frac{2.41845244 - 0.12734}{2.02} : θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Combine all the solutions

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = - arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn, θ = 2 π - arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Verify solutions by plugging them into the original equation

θ = arccos (- \frac{0.12734 + 2.41845244}{2.02}) + 2 πn, θ = arccos (\frac{2.41845244 - 0.12734}{2.02}) + 2 πn

Show solutions in decimal form

θ = 2.55514 \dots + 2 πn, θ = 0.78578 \dots + 2 πn

sec (x) + tan (x) = 3

cos (x) = \frac{10}{21}

cos (θ) = \frac{2}{3.3 8 ^{2} + 2 ^{2}}

3 tan (θ) - cot (θ) = 5

cos (x) \cdot sin (x) = 1

What is the general solution for sin(θ)=0.6367+0.1*cos(θ) ?
The general solution for sin(θ)=0.6367+0.1*cos(θ) is θ=2.55514…+2pin,θ=0.78578…+2pin