Answer
Verified
412.5k+ views
Hint: To solve this question first we will simplify the given expression by using trigonometric identities and formulas. We will find the value of x by simplifying the given expression. Then we will find the values of x in the given range.
Complete step by step answer:
We have been given an expression $\cos x+\sin x\tan x=2$.
We have to solve the given expression over the interval 0 to $2\pi $.
The given expression is $\cos x+\sin x\tan x=2$
Now, we know that $\tan x=\dfrac{\sin x}{\cos x}$
Substituting the value in the above expression we will get
$\Rightarrow \cos x+\sin x\dfrac{\sin x}{\cos x}=2$
Now, simplifying the above obtained equation we will get
$\Rightarrow \dfrac{{{\cos }^{2}}x+{{\sin }^{2}}x}{\cos x}=2$
Now, we know that ${{\cos }^{2}}x+{{\sin }^{2}}x=1$
Substituting the value in the above equation we will get
$\Rightarrow \dfrac{1}{\cos x}=2$
Now, simplifying the above obtained equation we will get
$\begin{align}
& \Rightarrow \dfrac{1}{2}=\cos x \\
& \Rightarrow \cos x=\dfrac{1}{2} \\
\end{align}$
Now, we have given the interval 0 to $2\pi $.
We know that cosine function has positive value in the first and fourth quadrant.
We know that $\cos \dfrac{\pi }{3}=\dfrac{1}{2}$
So we get $x=\dfrac{\pi }{3}$.
Now, take $x=\dfrac{\pi }{3}$ as reference angle to calculate the value of fourth quadrant angle we will get
$\Rightarrow 2\pi -\dfrac{\pi }{3}=\dfrac{5\pi }{3}$
Hence we get the value $x=\dfrac{\pi }{3},\dfrac{5\pi }{3}$ over the interval 0 to $2\pi $.
Note:
Students may consider all values of x for cosine function over the interval 0 to $2\pi $, which is incorrect. Here in this question when we solve the expression we get the positive value as $\dfrac{1}{2}$ so we need to consider only positive values of cosine function.
Complete step by step answer:
We have been given an expression $\cos x+\sin x\tan x=2$.
We have to solve the given expression over the interval 0 to $2\pi $.
The given expression is $\cos x+\sin x\tan x=2$
Now, we know that $\tan x=\dfrac{\sin x}{\cos x}$
Substituting the value in the above expression we will get
$\Rightarrow \cos x+\sin x\dfrac{\sin x}{\cos x}=2$
Now, simplifying the above obtained equation we will get
$\Rightarrow \dfrac{{{\cos }^{2}}x+{{\sin }^{2}}x}{\cos x}=2$
Now, we know that ${{\cos }^{2}}x+{{\sin }^{2}}x=1$
Substituting the value in the above equation we will get
$\Rightarrow \dfrac{1}{\cos x}=2$
Now, simplifying the above obtained equation we will get
$\begin{align}
& \Rightarrow \dfrac{1}{2}=\cos x \\
& \Rightarrow \cos x=\dfrac{1}{2} \\
\end{align}$
Now, we have given the interval 0 to $2\pi $.
We know that cosine function has positive value in the first and fourth quadrant.
We know that $\cos \dfrac{\pi }{3}=\dfrac{1}{2}$
So we get $x=\dfrac{\pi }{3}$.
Now, take $x=\dfrac{\pi }{3}$ as reference angle to calculate the value of fourth quadrant angle we will get
$\Rightarrow 2\pi -\dfrac{\pi }{3}=\dfrac{5\pi }{3}$
Hence we get the value $x=\dfrac{\pi }{3},\dfrac{5\pi }{3}$ over the interval 0 to $2\pi $.
Note:
Students may consider all values of x for cosine function over the interval 0 to $2\pi $, which is incorrect. Here in this question when we solve the expression we get the positive value as $\dfrac{1}{2}$ so we need to consider only positive values of cosine function.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE