Answer
Verified
414.9k+ views
Complete step-by-step solution
Figure:
Let us first draw a figure which represents the triangle and its angles A, B, and C, where \[\angle \] A is obtuse.
Since, we have been given that A is an obtuse angle, we will express it mathematically as,
\[\dfrac{\pi }{2} < A < \pi \] . . . . . . . . (1)
We know that A, B, C are angles of the same triangle. Therefore by the property Sum of angles of a triangle we get,
A+B+C = \[\pi \]
We need to get a range of values for B and C.
Angle A can be written as,
A = \[\pi - \left( {B + C} \right)\]
Let us substitute this in equation (1)
We get,
\[\dfrac{\pi }{2} < \pi - (B + C) < \pi \]
We want the angles B and C to be positive.
So we will multiply throughout the expression by” – “to keep the term (B+C) positive.
This will reverse the inequalities from “<” to “>”.
We get the following expression,
\[ - \dfrac{\pi }{2} > - \pi + (B + C) > - \pi \]
To simplify further we will add \[\pi \] throughout the expression. We get,
\[ \Rightarrow \dfrac{\pi }{2} > \left( {B + C} \right) > 0\]
Therefore, we now know that B+C is less than\[\dfrac{\pi }{2}\].
\[ \Rightarrow B + C < \dfrac{\pi }{2}\]
\[ \Rightarrow B < \dfrac{\pi }{2} - C\]
As we have to find the values of tanBtanC, we will apply tan on both sides of the inequality.
\[ \Rightarrow \tan (B) < \tan \left( {\dfrac{\pi }{2} - C} \right)\]
Using the trigonometric property, \[\tan \left( {\dfrac{\pi }{2} - \theta } \right) = \cot (\theta )\], we get
\[ \Rightarrow \tan B < \cot C\]
We also know that, \[\cot \theta = \dfrac{1}{{\tan \theta }}\]. Therefore we get,
\[ \Rightarrow \tan B < \dfrac{1}{{\tan C}}\]
We will now bring the terms $\tan B$ and $\tan C$ on the same side.
\[\therefore \tan B\tan C < 1\]
Hence, for the above-given question tanBtanC will be less than 1.
The correct answer choice is (C).
Note: We should keep in mind the two properties given in the question so that we keep the solution simple. Also, we should avoid opening the bracket initially as we might get confused. Keep in mind to keep (B+C) positive. We should also remember to get rid of the \[\pi \] in the middle term by adding the expression with \[\pi \].
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Why Are Noble Gases NonReactive class 11 chemistry CBSE
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
At which age domestication of animals started A Neolithic class 11 social science CBSE
Which are the Top 10 Largest Countries of the World?
Give 10 examples for herbs , shrubs , climbers , creepers
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
Write a letter to the principal requesting him to grant class 10 english CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE