Question

How do we find the value of \[{{\sin }^{2}}\theta \] if the value of \[1-{{\cos }^{2}}\theta \]= t ?

Answer 1

Hint: This question can be solved by using the trigonometric identities. You should know the formula that is the trigonometric identity of both sin and cos which is the \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]. After writing this, you just have to do one step to get the required answer. You just need to take cos to the right side and finally get the answer.

Complete step-by-step solution:
According to the problem, we are asked to find the value of \[{{\sin }^{2}}\theta \] if the value of \[1-{{\cos }^{2}}\theta \]= t .
For this, we should know the formula that is the trigonometric identity of both sin and cos which is the \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]. After writing this, we just need to take cos to the right side and finally get the answer.
Here we take the trigonometric identity \[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\] as 1.
\[{{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1\]------- (1)
Now we have to take the cos to the right side. After doing that we get:
\[\Rightarrow {{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta \] ------- (2)
Now as we can see from the question, we are given that \[1-{{\cos }^{2}}\theta =t\]. So substituting this in the equation 2, we get :
\[\Rightarrow {{\sin }^{2}}\theta =1-{{\cos }^{2}}\theta =t\]-
\[\Rightarrow {{\sin }^{2}}\theta =t\] ------ Final answer.
Here, we found out the value of \[{{\sin }^{2}}\theta \] as t.
Therefore, after all the solving, we get the final answer of the question - How do we find the value of \[{{\sin }^{2}}\theta \] if the value of \[1-{{\cos }^{2}}\theta \]= t as t.

Note: To solve this question, you should know the basic trigonometric identities. To solve this, we could also use the right angled triangle if you do not know the trigonometric identities.