Question

How do we write \[\tan \theta \] in terms of \[\sin \theta \]?

Answer 1

Hint: To solve this question first we assume a variable equal to \[\tan \theta \] function and then express that function in terms of \[\sin \theta \] and \[\cos \theta \]. Then we use trigonometry identity and express \[\cos \theta \] in terms of \[\sin \theta \] and then put that in the last relation and get the tan trigonometric function in terms of sin trigonometric function.

Complete step-by-step answer:
We have given a \[\tan \] trigonometric function and we have to write that in terms of \[\sin \] trigonometric function.
Let the given expression is equal to the variable \[x\].
\[x = \tan \theta \]
Now we can write \[\tan \theta \] in terms of \[\sin \theta \] and \[\cos \theta \].
We know that \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\]
\[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\] ……(i)
Now we express \[\cos \theta \] in terms of \[\sin \theta \].
We know the relation between \[\sin \theta \] and \[\cos \theta \] that is the identity of trigonometry.
The relation between \[\sin \theta \] and \[\cos \theta \] is :- \[{\sin ^2}\theta + {\cos ^2}\theta = 1\]
 Now we express \[\cos \theta \] in terms of \[\sin \theta \]
\[{\sin ^2}\theta + {\cos ^2}\theta = 1\]
On rearranging the equation.
\[{\cos ^2}\theta = 1 - {\sin ^2}\theta \]
On taking root both side-
\[\cos \theta = \sqrt {1 - {{\sin }^2}\theta } \]
Now we put this relation in the equation (i)
\[\tan \theta = \dfrac{{\sin \theta }}{{\sqrt {1 - {{\sin }^2}\theta } }}\]
Here we are able to express tan trigonometric function in terms of sin trigonometric function.
Final answer:
we write \[\tan \theta \] in terms of \[\sin \theta \] such as
\[ \Rightarrow \tan \theta = \dfrac{{\sin \theta }}{{\sqrt {1 - {{\sin }^2}\theta } }}\]

Note: To solve these types of questions we must know all the relations between the trigonometric function, formulas of trigonometry, and the identities. If we have to ask in terms of \[\cos \theta \] then we have to replace \[\sin \theta \] in term so \[\cos \theta \] and if they ask us to write in terms of sec and cosec then we change sin and cos according. This question is very easy but you must know all the basics of trigonometry.