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Question: If \( \cos \theta = \left( {\dfrac{4}{5}} \right) \) , where \( 0 \leqslant \theta \leqslant \left( ...

If cos⁡θ=(45)\cos \theta = \left( {\dfrac{4}{5}} \right) , where 0⩽θ⩽(π2)0 \leqslant \theta \leqslant \left( {\dfrac{\pi }{2}} \right) . How do you find the value of sin⁡θ\sin \theta ?

Explanation

Solution

Hint : In the given problem, we are required to calculate sine of an angle whose cosine is given to us beforehand in the question itself. Such problems require basic knowledge of trigonometric ratios and formulae. Besides this, knowledge of concepts of algebraic rules and properties is extremely essential to answer these questions correctly.

Complete step-by-step answer :
In the given problem, we are required to find the sine of an angle whose cosine is given to us. So, we should remember trigonometric identity involving both sine and cosine functions: sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1 .
So, cos⁡θ=(45)\cos \theta = \left( {\dfrac{4}{5}} \right) .
Now, using the trigonometric identity sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1 , we get,
⇒sin⁡2θ+(45)2=1\Rightarrow {\sin ^2}\theta + {\left( {\dfrac{4}{5}} \right)^2} = 1
⇒sin⁡2θ=1−1625\Rightarrow {\sin ^2}\theta = 1 - \dfrac{{16}}{{25}}
⇒sin⁡2θ=25−1616\Rightarrow {\sin ^2}\theta = \dfrac{{25 - 16}}{{16}}
⇒sin⁡2θ=916\Rightarrow {\sin ^2}\theta = \dfrac{9}{{16}}
Now, we have to take the square root on both sides of the equation.
Now, we know that cosine function is positive in the first quadrant and sine function is also positive in the first quadrant.
⇒sin⁡θ=916\Rightarrow \sin \theta = \sqrt {\dfrac{9}{{16}}}
We know that the square root of 1616 is 44 . So, we get,
⇒sin⁡θ=34\Rightarrow \sin \theta = \dfrac{3}{4}
So, the value of sine of angle θ\theta is (34)\left( {\dfrac{3}{4}} \right) .
So, the correct answer is “ (34)\left( {\dfrac{3}{4}} \right) ”.

Note : For finding the value of a trigonometric function for an angle given any other trigonometric ratio, we can use trigonometric identities. Then we find the required trigonometric ratio with help of basic trigonometric formulae and definitions of trigonometric ratios. Such questions require clarity of basic concepts of trigonometric functions as well as their algebraic rules like transposition.