solve-for-0-360-the-equation-cos-pi-3-1-2-

Question Number 36092 by Rio Mike last updated on 28/May/18

solve for 0 ° ⩽ θ ⩽ 360 ° the equation

\cos (θ + \frac{π}{3}) = \frac{1}{2}

Commented by abdo mathsup 649 cc last updated on 30/May/18

(e) \Leftrightarrow c o s (θ + \frac{π}{3}) = c o s (\frac{π}{3}) \Leftrightarrow θ + \frac{π}{3} = \frac{π}{3} + 2 k π

o r θ + \frac{π}{3} = - \frac{π}{3} + 2 k π \Leftrightarrow θ = 2 k π o r

θ = - \frac{2 π}{3} + 2 k π (k \in Z)

c a s e 1 0 ⩽ 2 k π ⩽ 2 π \Rightarrow 0 ⩽ k ⩽ 1 \Rightarrow k = 0 o r k = 1

c a s e 2 0 ⩽ - \frac{2 π}{3} + 2 k π ⩽ 2 π \Rightarrow

0 ⩽ - \frac{2}{3} + 2 k ⩽ 2 \Rightarrow 0 ⩽ - \frac{1}{3} + k ⩽ 1 \Rightarrow

\frac{1}{3} ⩽ k ⩽ \frac{4}{3} \Rightarrow k = 1 s o t h e s o l u t i o n f o r t h i s

e q u a t i o n a r e 0, 2 π, \frac{4 π}{3} .

Answered by Joel579 last updated on 28/May/18

\cos (θ + \frac{π}{3}) = \cos (\frac{π}{3})

θ + \frac{π}{3} = \frac{π}{3}

θ = 0

\cos (θ + \frac{π}{3}) = \cos (\frac{5 π}{3})

θ + \frac{π}{3} = \frac{5 π}{3}

θ = \frac{4 π}{3}

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