Answer

Verified

412.5k+ views

**Hint:**In question it is asked that, we have to find the value of $\sin \theta $ given that $\dfrac{\sec \theta -\tan \theta }{\sec \theta +\tan \theta }=\dfrac{1}{4}$.

So, to do so we will use identities and properties of trigonometric ratios such as $\tan \theta =\dfrac{\sin \theta }{\cos \theta }$ and $\sec \theta =\dfrac{1}{\cos \theta }$ so as to obtain the $\sin \theta $.

**Complete step-by-step solution:**We know that \[sin\text{ }\theta \text{ },\text{ }cos\text{ }\theta \text{ },\text{ }tan\text{ }\theta \text{ },\text{ }cot\text{ }\theta \text{ },\text{ }sec\text{ }\theta \text{ }and\text{ }cosec\text{ }\theta \] are trigonometric function, where \[\theta \] is the angle made by the hypotenuse with the base of triangle.

Now, in question it is given that $\dfrac{\sec \theta -\tan \theta }{\sec \theta +\tan \theta }=\dfrac{1}{4}$.

Now, also we know that tan A equal to the ratio of the sine function and cos A function that is $\tan A=\dfrac{\sin A}{\cos A}$.

And, also $sec\theta $ is equals to reciprocal of trigonometric function $\cos \theta $ that is $\sec \theta =\dfrac{1}{\cos \theta }$.

so, we can write $\dfrac{\sec \theta -\tan \theta }{\sec \theta +\tan \theta }=\dfrac{1}{4}$ as,

$\dfrac{\dfrac{1}{\cos \theta }-\dfrac{\sin A}{\cos A}}{\dfrac{1}{\cos \theta }+\dfrac{\sin A}{\cos A}}=\dfrac{1}{4}$.

On taking L.C.M in numerator an denominator, we get

$\dfrac{\dfrac{1-\sin \theta }{\cos \theta }}{\dfrac{1+\sin \theta }{\cos \theta }}=\dfrac{1}{4}$

Taking 4 from the denominator of right hand side to numerator of left hand side, $\dfrac{1+\sin \theta }{\cos \theta }$ from numerator of right hand side of left hand side to denominator of right hand side using cross multiplication, we get

$4\cdot \left( \dfrac{1-\sin \theta }{\cos \theta } \right)=\dfrac{1+\sin \theta }{\cos \theta }$……………........( i )

On solving brackets,

$\dfrac{4-4\sin \theta }{\cos \theta }=\dfrac{1+\sin \theta }{\cos \theta }$

On simplifying equation, we get

$4-4\sin \theta =1+\sin \theta $

Taking, $4\sin \theta $ from left hand side to right hand side and 1 from right hand side to left hand side,

$4-1=\sin \theta +4\sin \theta $

On solving, we get

$3=5\sin \theta $

Taking, 4 from the numerator of right hand side to denominator of left hand side, we get

$\sin \theta =\dfrac{3}{5}$

**Hence, if $\dfrac{\sec \theta -\tan \theta }{\sec \theta +\tan \theta }=\dfrac{1}{4}$ , then the value of $\sin \theta $ is equals to $\dfrac{3}{5}$.**

**Note:**One must know all trigonometric identities, properties, and relationships between trigonometric functions. While solving the question always use the most appropriate substitution of trigonometric relation which directly leads to the result. There may be calculation mistakes in cross multiplication, so be careful while solving an expression.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE