Question

What is the exact value of $\sin 65^\circ ?$

Answer 1

Hint: As we know that the above-given function is a trigonometric function. These are some of the basic trigonometric functions such as sine, cosine, tangent, secant, cotangent and cosecant.
Here we have to find the value of $\sin 65$ and since it is not a standard angle, we will try to find the value in radians. We should know that the value of $\sin 65$ in degrees is the same as the value of $\sin 65$ in radians. Therefore we will apply their formula and solve it.

Complete step-by-step answer:
According to the given question we have to find the trigonometric value, $\sin 65^\circ $.
To find the value of trigonometric degree in radians, we have to multiply the value with
$\dfrac{\pi }{{180^\circ }}$ ,
Therefore, by applying this we can write:
 $ = 65^\circ \times \dfrac{\pi }{{180^\circ }}$ ,
On simplifying the value we can write this also as
$ = \dfrac{{13\pi }}{{36}}$ .
This is the value that we get which is also in decimal as $1.134466...$
There is no such exact value of $\sin 65^\circ $
Hence this is our required answer.

Note: We should note that we can cross check our answer by converting radians into degrees. We can convert radian back into degrees by multiplying it with $\dfrac{{180}}{\pi }$ . We should also know the trigonometric formula such as
 $\sin (C + D) = \sin C \cdot \operatorname{Sin} D + \cos C \cdot \cos D$
So, if we have to find the value of $\sin 75$ then we can break the angle of sine into a sum of two standard angles.
We can write $\sin {75^ \circ } = \sin ({45^ \circ } + {30^ \circ })$.
We should also know that the inverse trigonometric functions are also known as arcus function, anti-trigonometric function or cyclomatic function. These basic inverse trigonometric functions are used to find the missing angles in right triangles.