What is the general solution for 2cos(t)=1+sin(t) ?

The general solution for 2cos(t)=1+sin(t) is t=(3pi)/2+2pin,t=0.64350…+2pin

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2cos(t)=1+sin(t)

2 cos (t) = 1 + sin (t)

t = \frac{3 π}{2} + 2 πn, t = 0.64350 \dots + 2 πn

Solution steps

2 cos (t) = 1 + sin (t)

Square both sides

(2 cos (t))^{2} = (1 + sin (t))^{2}

Subtract

(1 + sin (t))^{2}

from both sides

4 cos^{2} (t) - 1 - 2 sin (t) - sin^{2} (t) = 0

Rewrite using trig identities

3 - 2 sin (t) - 5 sin^{2} (t) = 0

Solve by substitution

sin (t) = - 1, sin (t) = \frac{3}{5}

sin (t) = - 1 : t = \frac{3 π}{2} + 2 πn

sin (t) = \frac{3}{5} : t = arcsin (\frac{3}{5}) + 2 πn, t = π - arcsin (\frac{3}{5}) + 2 πn

Combine all the solutions

t = \frac{3 π}{2} + 2 πn, t = arcsin (\frac{3}{5}) + 2 πn, t = π - arcsin (\frac{3}{5}) + 2 πn

Verify solutions by plugging them into the original equation

t = \frac{3 π}{2} + 2 πn, t = arcsin (\frac{3}{5}) + 2 πn

Show solutions in decimal form

t = \frac{3 π}{2} + 2 πn, t = 0.64350 \dots + 2 πn

cos (θ) = - \frac{5}{6}, csc (θ)

sin^{2} (x) = - 2 + 3 sin (x) + cos^{2} (x)

sin (x) - 3 cos (x) = - 1

sin (θ) = \frac{- 1}{17}

tan (x) + cot (x) = 3, = tan (3 x) + cot (3 x)

What is the general solution for 2cos(t)=1+sin(t) ?
The general solution for 2cos(t)=1+sin(t) is t=(3pi)/2+2pin,t=0.64350…+2pin