What is the general solution for sin(θ)-(cos(θ))/5 =0.6377 ?

The general solution for sin(θ)-(cos(θ))/5 =0.6377 is θ=2.66345…+2pin,θ=0.87293…+2pin

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sin(θ)-(cos(θ))/5 =0.6377

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

θ = 2.66345 \dots + 2 πn, θ = 0.87293 \dots + 2 πn

Solution steps

sin (θ) - \frac{cos ( θ )}{5} = 0.6377

Add

\frac{cos ( θ )}{5}

to both sides

sin (θ) = 0.6377 + 0.2 cos (θ)

Square both sides

sin^{2} (θ) = (0.6377 + 0.2 cos (θ))^{2}

Subtract

(0.6377 + 0.2 cos (θ))^{2}

from both sides

sin^{2} (θ) - 0.40666129 - 0.25508 cos (θ) - 0.04 cos^{2} (θ) = 0

Rewrite using trig identities

0.59333871 - 0.25508 cos (θ) - 1.04 cos^{2} (θ) = 0

Solve by substitution

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08}, cos (θ) = \frac{2.53335484 - 0.25508}{2.08}

cos (θ) = - \frac{0.25508 + 2.53335484}{2.08} : θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = - arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn

cos (θ) = \frac{2.53335484 - 0.25508}{2.08} : θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

Combine all the solutions

θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = - arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn, θ = 2 π - arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

Verify solutions by plugging them into the original equation

θ = arccos (- \frac{0.25508 + 2.53335484}{2.08}) + 2 πn, θ = arccos (\frac{2.53335484 - 0.25508}{2.08}) + 2 πn

Show solutions in decimal form

θ = 2.66345 \dots + 2 πn, θ = 0.87293 \dots + 2 πn

cos (2 θ) = 0.4, 0^{\circ} \leq θ \leq 36 0^{\circ}

sin (4 x) = cos (4 x)

2 cos^{2} (x) + 9 cos (x) + 4 = 0

x, arccsc (2 x) = \frac{π}{6}

sin^{2} (x) + tan^{2} (x) = 1

What is the general solution for sin(θ)-(cos(θ))/5 =0.6377 ?
The general solution for sin(θ)-(cos(θ))/5 =0.6377 is θ=2.66345…+2pin,θ=0.87293…+2pin