Question

How do you find the exact value of \[\sin 45^\circ \]?

Answer 1

Hint:
In the given question, we have been asked the exact value of a trigonometric ratio with a constant angle. By exact value, it means that if the value is in fractions, we have to convert it to decimals. This is achieved by dividing the numerator by the denominator. If the denominator is an irrational number, then we shift the irrational number to the numerator by rationalizing it.

Complete Step by Step Solution:
We know, \[\sin 45^\circ = \dfrac{1}{{\sqrt 2 }}\]
The given number has the denominator which is an irrational number. So first, we rationalize it,
\[\dfrac{1}{{\sqrt 2 }} = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = \dfrac{{\sqrt 2 }}{2}\]
Now, \[\sqrt 2 = 1.414\]
So, \[\dfrac{{\sqrt 2 }}{2} = \dfrac{{1.414}}{2} = 0.707\]

Hence, the exact value of \[\sin 45^\circ \] is \[0.707\].

Note:
For solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we write the formula which connects the two things. We have to give care when we are rationalizing the denominator as it is the only place where we could make an error.