# Find the general solution of the equation $\cos \theta =-\dfrac{1}{2}$ is

A. $\theta =n\pi \pm \dfrac{2\pi }{3},n\in \mathbb{Z}$

B. $\theta =2n\pi \pm \dfrac{2\pi }{3},n\in \mathbb{Z}$

C. $\theta =n\pi \pm \dfrac{\pi }{3},n\in \mathbb{Z}$

D. none of these

Answer

Verified

277.5k+ views

**Hint:**We first find the principal value of x for which $\cos \theta =-\dfrac{1}{2}$. In that domain, equal value of the same ratio gives equal angles. We find the angle value for $\theta $. At the end we also find the general solution for the equation $\cos \theta =-\dfrac{1}{2}$.

**Complete step by step solution:**

It’s given that $\cos \theta =-\dfrac{1}{2}$. The value in fraction is $-\dfrac{1}{2}$. We need to find $\theta $ for which $\cos \theta =-\dfrac{1}{2}$.

We know that in the principal domain or the periodic value of $0\le x\le \pi $ for $\cos x$, if we get $\cos a=\cos b$ where $0\le a,b\le \pi $ then $a=b$.

We have the value of $\cos \left( \dfrac{2\pi }{3} \right)$ as $-\dfrac{1}{2}$. $0<\dfrac{2\pi }{3}<\pi $.

Therefore, \[\cos \theta =-\dfrac{1}{2}=\cos \left( \dfrac{2\pi }{3} \right)\] which gives \[\theta =\dfrac{2\pi }{3}\].

We need to find the general solution then the domain changes to $-\infty \le x\le \infty $. In that case we have to use the formula $x=2n\pi \pm a$ for $\cos \left( x \right)=\cos a$ where $0\le x\le \pi $. For our given problem $\cos \theta =-\dfrac{1}{2}$, the general solution will be $\theta =2n\pi \pm \dfrac{2\pi }{3}$. Here $n\in \mathbb{Z}$.

**Note:**

We also can show the solutions (primary and general) of the equation $\cos \theta =-\dfrac{1}{2}$ through a graph. We take $y=\cos \theta =-\dfrac{1}{2}$. We got two equations: $y=\cos \theta $ and $y=-\left( \dfrac{1}{2} \right)$. We place them on the graph and find the solutions as their intersecting points.

We can see the primary solution in the interval $0\le \theta \le \pi $ is the point A as $\theta =\dfrac{2\pi }{3}$.

All the other intersecting points of the curve and the line are general solutions.

Recently Updated Pages

Which of the following would not be a valid reason class 11 biology CBSE

What is meant by monosporic development of female class 11 biology CBSE

Draw labelled diagram of the following i Gram seed class 11 biology CBSE

Explain with the suitable examples the different types class 11 biology CBSE

How is pinnately compound leaf different from palmately class 11 biology CBSE

Match the following Column I Column I A Chlamydomonas class 11 biology CBSE

Trending doubts

The lightest gas is A nitrogen B helium C oxygen D class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Which place is known as the tea garden of India class 8 social science CBSE

What is pollution? How many types of pollution? Define it

Write a letter to the principal requesting him to grant class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a diagram of a plant cell and label at least eight class 11 biology CBSE