Answer
384.6k+ views
Hint: First express $2$ in terms of tangent of an argument and find the value of respective argument then convert tangent into sine and cosine and simplify further using the compound angle formula of sine and finally find the required solution in the principal interval. Compound angle formula of sine function is given as:
$\sin (a - b) = \sin a\cos b - \sin b\cos a$
Use the above information to solve the problem.
Formula used:
Compound angle formula of sine, $\sin (a - b) = \sin a\cos b - \sin b\cos a$
Complete step by step solution:
In order to solve the given trigonometric equation $\sin x - 2\cos x = 2$
we will first express the coefficient of cosine in terms of tangent,
That is let us consider,
$
\tan y = 2 \\
\therefore y = {\tan ^{ - 1}}(2) = {63.43^0} \\
$
We can write the given trigonometric equation as
$
\Rightarrow \sin x - 2\cos x = 2 \\
\Rightarrow \sin x - \tan y\cos x = 2 \\
$
Now expressing tangent as the ratio of sine is to cosine, we will get
$
\Rightarrow \sin x - \dfrac{{\sin y}}{{\cos y}}\cos x = 2 \\
\Rightarrow \sin x\cos y - \sin y\cos x = 2\cos y \\
$
Using compound angle formula of sine function to solve it further, we will get
$
\Rightarrow \sin x\cos y - \sin y\cos x = 2\cos y \\
\Rightarrow \sin (x - y) = 2\cos y \\
$
Now putting the value of $y$ and also substituting the value of $\cos y$ we will get,
$
\Rightarrow \sin \left( {x - {{63.43}^0}} \right) = 2\cos \left( {{{63.43}^0}} \right) \\
\Rightarrow \sin \left( {x - {{63.43}^0}} \right) = 2 \times 0.45\;\;\;\;\;\left[ {\because \cos \left(
{{{63.43}^0}} \right) = 0.45} \right] \\
\Rightarrow \sin \left( {x - {{63.43}^0}} \right) = 0.90 \\
$
Taking inverse function of sine both sides we will get,
$
\Rightarrow {\sin ^{ - 1}}\left( {\sin \left( {x - {{63.43}^0}} \right)} \right) = {\sin ^{ - 1}}\left( {0.90}
\right) \\
\Rightarrow x - {63.43^0} = {\sin ^{ - 1}}\left( {0.90} \right) \\
$
Now, ${\sin ^{ - 1}}\left( {0.90} \right)$ will give two values in the principle interval i.e. $\left(
{0,\;{{360}^0}} \right)$
One of the two solutions is given as
$
\Rightarrow x - {63.43^0} = {\sin ^{ - 1}}\left( {0.90} \right) \\
\Rightarrow x - {63.43^0} = {64.16^0} \\
\Rightarrow x = {63.43^0} + {64.16^0} \approx {128^0} \\
$
Another one is given as
\[
\Rightarrow x - {63.43^0} = {\sin ^{ - 1}}\left( {0.90} \right) \\
\Rightarrow x - {63.43^0} = {180^0} - {64.16^0} \\
\Rightarrow x = {115.84^0} + {63.43^0} = {179.27^0} \\
\]
Therefore we get two solutions for the trigonometric equation $\sin x - 2\cos x = 2$ in the principle interval.
Note: We have taken both the solutions, because the sine function shows positive sign twice in between the interval $\left( {0,\;{{360}^0}} \right)$ that is in the first quadrant and in the second quadrant. You can check both the solutions by substituting it in the function. Also, when finding the inverse value of a trigonometric function then always considers its principle solution.
$\sin (a - b) = \sin a\cos b - \sin b\cos a$
Use the above information to solve the problem.
Formula used:
Compound angle formula of sine, $\sin (a - b) = \sin a\cos b - \sin b\cos a$
Complete step by step solution:
In order to solve the given trigonometric equation $\sin x - 2\cos x = 2$
we will first express the coefficient of cosine in terms of tangent,
That is let us consider,
$
\tan y = 2 \\
\therefore y = {\tan ^{ - 1}}(2) = {63.43^0} \\
$
We can write the given trigonometric equation as
$
\Rightarrow \sin x - 2\cos x = 2 \\
\Rightarrow \sin x - \tan y\cos x = 2 \\
$
Now expressing tangent as the ratio of sine is to cosine, we will get
$
\Rightarrow \sin x - \dfrac{{\sin y}}{{\cos y}}\cos x = 2 \\
\Rightarrow \sin x\cos y - \sin y\cos x = 2\cos y \\
$
Using compound angle formula of sine function to solve it further, we will get
$
\Rightarrow \sin x\cos y - \sin y\cos x = 2\cos y \\
\Rightarrow \sin (x - y) = 2\cos y \\
$
Now putting the value of $y$ and also substituting the value of $\cos y$ we will get,
$
\Rightarrow \sin \left( {x - {{63.43}^0}} \right) = 2\cos \left( {{{63.43}^0}} \right) \\
\Rightarrow \sin \left( {x - {{63.43}^0}} \right) = 2 \times 0.45\;\;\;\;\;\left[ {\because \cos \left(
{{{63.43}^0}} \right) = 0.45} \right] \\
\Rightarrow \sin \left( {x - {{63.43}^0}} \right) = 0.90 \\
$
Taking inverse function of sine both sides we will get,
$
\Rightarrow {\sin ^{ - 1}}\left( {\sin \left( {x - {{63.43}^0}} \right)} \right) = {\sin ^{ - 1}}\left( {0.90}
\right) \\
\Rightarrow x - {63.43^0} = {\sin ^{ - 1}}\left( {0.90} \right) \\
$
Now, ${\sin ^{ - 1}}\left( {0.90} \right)$ will give two values in the principle interval i.e. $\left(
{0,\;{{360}^0}} \right)$
One of the two solutions is given as
$
\Rightarrow x - {63.43^0} = {\sin ^{ - 1}}\left( {0.90} \right) \\
\Rightarrow x - {63.43^0} = {64.16^0} \\
\Rightarrow x = {63.43^0} + {64.16^0} \approx {128^0} \\
$
Another one is given as
\[
\Rightarrow x - {63.43^0} = {\sin ^{ - 1}}\left( {0.90} \right) \\
\Rightarrow x - {63.43^0} = {180^0} - {64.16^0} \\
\Rightarrow x = {115.84^0} + {63.43^0} = {179.27^0} \\
\]
Therefore we get two solutions for the trigonometric equation $\sin x - 2\cos x = 2$ in the principle interval.
Note: We have taken both the solutions, because the sine function shows positive sign twice in between the interval $\left( {0,\;{{360}^0}} \right)$ that is in the first quadrant and in the second quadrant. You can check both the solutions by substituting it in the function. Also, when finding the inverse value of a trigonometric function then always considers its principle solution.
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)