Answer
384.6k+ views
Hint: Here we will use the trigonometric identity and will simplify the equation using it. Also, will use the All STC law to find the angles in the interval between $0^\circ $to $360^\circ $.
Complete step-by-step solution:
Here we will use the identity for $\cos 2x = 1 - 2{\sin ^2}x$
Place the above value in the given equation.
$1 - 2{\sin ^2}x - \sin x = 1$
Like terms with the same value and same sign on the opposite sides of the equations cancel each other.
$ - 2{\sin ^2}x - \sin x = 0$
Take the negative sign common from the above equation.
$2{\sin ^2}x + \sin x = 0$
Take the common factor from the above equation from both the terms in it.
$\sin x(2\sin x + 1) = 0$
Hence, the roots of the equation will be either
$\sin x = 0$ or $2\sin x + 1 = 0$
First solve for $\sin x = 0$
Referring to the trigonometric table for values,
$ \Rightarrow x = 0^\circ ,180^\circ $ ….. (A)
Now, $2\sin x + 1 = 0$
Take constant on the right hand side of the equation, when you move any term from one side of the equation to other then the sign of the term also changes. Positive terms become negative and vice versa.
$2\sin x = - 1$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$\sin x = - \frac{1}{2}$
Using All STC rule, sine function is negative in the third and the fourth quadrant.
$ \Rightarrow x = 210^\circ ,330^\circ $ ….. (B)
Hence, the resultant answer is
$x = 0^\circ ,180^\circ $
or
$x = 210^\circ ,330^\circ $
Note: Remember the All STC rule, it is also known as ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ $ ).
Complete step-by-step solution:
Here we will use the identity for $\cos 2x = 1 - 2{\sin ^2}x$
Place the above value in the given equation.
$1 - 2{\sin ^2}x - \sin x = 1$
Like terms with the same value and same sign on the opposite sides of the equations cancel each other.
$ - 2{\sin ^2}x - \sin x = 0$
Take the negative sign common from the above equation.
$2{\sin ^2}x + \sin x = 0$
Take the common factor from the above equation from both the terms in it.
$\sin x(2\sin x + 1) = 0$
Hence, the roots of the equation will be either
$\sin x = 0$ or $2\sin x + 1 = 0$
First solve for $\sin x = 0$
Referring to the trigonometric table for values,
$ \Rightarrow x = 0^\circ ,180^\circ $ ….. (A)
Now, $2\sin x + 1 = 0$
Take constant on the right hand side of the equation, when you move any term from one side of the equation to other then the sign of the term also changes. Positive terms become negative and vice versa.
$2\sin x = - 1$
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
$\sin x = - \frac{1}{2}$
Using All STC rule, sine function is negative in the third and the fourth quadrant.
$ \Rightarrow x = 210^\circ ,330^\circ $ ….. (B)
Hence, the resultant answer is
$x = 0^\circ ,180^\circ $
or
$x = 210^\circ ,330^\circ $
Note: Remember the All STC rule, it is also known as ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ($0^\circ \;{\text{to 90}}^\circ $ ) are positive, sine and cosec are positive in the second quadrant ($90^\circ {\text{ to 180}}^\circ $ ), tan and cot are positive in the third quadrant ($180^\circ \;{\text{to 270}}^\circ $ ) and sin and cosec are positive in the fourth quadrant ($270^\circ {\text{ to 360}}^\circ $ ).
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)