Question

If the value of ${\cos ^2}\theta = \dfrac{4}{5}$, then ${\sin ^2}\theta $ =
$(a){\text{ }}\dfrac{1}{5}$
$(b){\text{ }}\dfrac{2}{5}$
$(c){\text{ }}\dfrac{{ - 1}}{5}$
$(d){\text{ }}\dfrac{3}{5}$

Answer 1

Hint: To solve such types of trigonometric problems, simply remember the various trigonometric identities and substitute the given values to find the values of the unknown trigonometric functions.

Complete step-by-step answer:
We have the given trigonometric value as

${\cos ^2}\theta = \dfrac{4}{5}$ … (1)

Now, we know the trigonometric identity

${\sin ^2}\theta + {\cos ^2}\theta = 1$

By substituting the equation (1) in the above given identity, we get the equation as

${\sin ^2}\theta + \dfrac{4}{5} = 1$

$ \Rightarrow {\sin ^2}\theta = 1 - \dfrac{4}{5}$

$ \Rightarrow {\sin ^2}\theta = \dfrac{{5 - 4}}{5}$

$\therefore {\sin ^2}\theta = \dfrac{1}{5}$

Hence, the correct solution is the option$(a){\text{ }}\dfrac{1}{5}$.

Note: The above given question can be solved by the use of some other trigonometric identities as well. This is not mandatory to use the same identities used above. But this will require an adequate knowledge about the properties of the trigonometric functions and their respective identities.