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Question: If \(\cos\theta = \frac{- 1}{2}\text{and}0{^\circ} < \theta < \text{360}{^\circ},\)then the values o...

If cos⁡θ=−12and0∘<θ<360∘,\cos\theta = \frac{- 1}{2}\text{and}0{^\circ} < \theta < \text{360}{^\circ},then the values ofθ\thetaare

A

120∘and 300∘120{^\circ}\text{and }300{^\circ}

B

60∘and 120∘60{^\circ}\text{and }120{^\circ}

C

120∘and 240∘120{^\circ}\text{and }240{^\circ}

D

60∘and 240∘60{^\circ}\text{and }240{^\circ}

Answer

120∘and 240∘120{^\circ}\text{and }240{^\circ}

Explanation

Solution

Given, cos⁡θ=−12\cos\theta = - \frac{1}{2} and 0∘<θ<360∘.0{^\circ} < \theta < 360{^\circ}. We know that

cos⁡60∘=12\cos 60{^\circ} = \frac{1}{2} and cos⁡(180∘−60∘)=−cos⁡60∘=−12\cos(180{^\circ} - 60{^\circ}) = - \cos 60{^\circ} = \frac{- 1}{2} or

cos⁡120∘=−12.\cos 120{^\circ} = - \frac{1}{2}. Similarly cos⁡(180o+60o)=−cos⁡60∘=−12\cos(180^{o} + 60^{o}) = - \cos 60{^\circ} = - \frac{1}{2}

or cos⁡240∘=−12.\cos 240{^\circ} = \frac{- 1}{2}.

Therefore θ=120∘\theta = 120{^\circ} and 240°.