Question

Find \[\dfrac{{\tan 41^\circ }}{{\cos 49^\circ }}\]

Answer 1

Hint:
Here, we will use trigonometric identities to convert the denominator into sine function. We will then convert the tangent function in terms of sine and cosine function. Then we will cancel out the like trigonometric functions and solve it further to get the required value.

Formula Used:
We will use the following formulas:
1) \[\cos \left( {90^\circ - \theta } \right) = \sin \theta \]
2) \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
3) \[\sec \theta = \dfrac{1}{{\cos \theta }}\]

Complete Step by step Solution:
We have to find: \[\dfrac{{\tan 41^\circ }}{{\cos 49^\circ }}\]
Now, we know the identity \[\cos \left( {90^\circ - \theta } \right) = \sin \theta \].
Here, in the denominator, applying this formula and writing the angle of cosine in terms of the difference of \[\left( {90^\circ - \theta } \right)\], we get,
\[\dfrac{{\tan 41^\circ }}{{\cos 49^\circ }} = \dfrac{{\tan 41^\circ }}{{\cos \left( {90^\circ - 41^\circ } \right)}}\]
\[ \Rightarrow \dfrac{{\tan 41^\circ }}{{\cos 49^\circ }} = \dfrac{{\tan 41^\circ }}{{\sin 41^\circ }}\]
Substituting \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\] in the above equation, we get
\[ \Rightarrow \dfrac{{\tan 41^\circ }}{{\cos 49^\circ }} = \dfrac{{\sin 41^\circ }}{{\cos 41^\circ \times \sin 41^\circ }}\]
Cancelling the same trigonometric terms from the numerator and denominator, we get
\[ \Rightarrow \dfrac{{\tan 41^\circ }}{{\cos 49^\circ }} = \dfrac{1}{{\cos 41^\circ }}\]
We know that \[\sec \theta = \dfrac{1}{{\cos \theta }}\].
Hence, we can also write our answer as:
\[ \Rightarrow \dfrac{{\tan 41^\circ }}{{\cos 49^\circ }} = \sec 41^\circ \]
Therefore, the value of \[\dfrac{{\tan 41^\circ }}{{\cos 49^\circ }} = \sec 41^\circ \]

Hence, this is the required answer.

Note:
Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers to make maps. It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In the simple terms they are written as ‘sin’, ‘cos’ and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.