Question

If $\sin \theta = \dfrac{3}{5}$ and $\theta $ is acute then find the value of $\dfrac{{\tan \theta - 2\cos \theta }}{{3\sin \theta + \sec \theta }}$.

Answer 1

Hint: By using the basic trigonometric identity given below we can simplify the above expression that is $\dfrac{{\tan \theta - 2\cos \theta }}{{3\sin \theta + \sec \theta }}$ . In order to solve and simplify the given expression we have to use the identities and express our given expression in the simplest form and thereby solve it. . Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily.

Complete step-by-step solution:
It is given that , $\sin \theta = \dfrac{3}{5}$ , and
It is also given that $\theta $ is an acute angle.
Then, we can say that the opposite side of the right-angled triangle containing angle $\theta $ is of length three and hypotenuse is of length five ,
Now, we have to Use Pythagoras theorem, we get the adjacent side $ = 4$
Now we have, $\cos \theta = \dfrac{{Adjacent}}{{Hypotenuse}} = \dfrac{4}{5}$
And \[\tan \theta = \dfrac{{Opposite}}{{Adjacent}} = \dfrac{3}{4}\]
So, $\dfrac{{\tan \theta - 2\cos \theta }}{{3\sin \theta + \sec \theta }}$ $ = \dfrac{{\dfrac{3}{4} - 2\left( {\dfrac{4}{5}} \right)}}{{3\left( {\dfrac{3}{5}} \right) + \dfrac{5}{4}}} = - \dfrac{{17}}{{61}}$

Note: Some other equations needed for solving these types of problem are:
$\cos \theta = \dfrac{{Adjacent}}{{Hypotenuse}}$ and
\[\tan \theta = \dfrac{{Opposite}}{{Adjacent}}\] .
Range of cosine and sine: $\left[ { - 1,1} \right]$ ,

Also, while approaching a trigonometric problem one should keep in mind that one should work with one side at a time and manipulate it to the other side. The most straightforward way to do this is to simplify one side to the other directly, but we can also transform both sides to a common expression if we see no direct way to connect the two. Questions similar in nature as that of above can be approached in a similar manner and we can solve it easily.