Question

How do you find the exact value of $\sin 7.5?$

Answer 1

Hint : As we know that the above given function is a trigonometric function i.e. sine function. Sine of an angle is defined as the ratio of the side opposite to the angle to the hypotenuse in a right angle triangle. We know that the half angle identities for the sine function are derived from the two of the cosine identities, $\sin \dfrac{\theta }{2} = \pm \dfrac{{\sqrt {1 - \cos \theta } }}{2}$ . The positive or negative sign of the value depends on the quadrant in which the resulting angle is located.

Complete step-by-step answer :
formula to solve it. We know that
$\cos \dfrac{\theta }{2} = \pm \dfrac{{\sqrt {1 + \cos \theta } }}{2}$ .
Also we know that
$\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$ , so by applying $\theta = 30$
We have,
$\cos \dfrac{{{{30}^ \circ }}}{2} = \sqrt {\dfrac{{1 + \cos {{30}^ \circ }}}{2}}$
$\Rightarrow \cos {15^ \circ } = \sqrt {\dfrac{{1 + \dfrac{{\sqrt 3 }}{2}}}{2}}$
Which gives
$\cos {15^ \circ } = \sqrt {\dfrac{{2 + \sqrt 3 }}{4}}$ .
Therefore the value of $\cos 15 = \dfrac{{\sqrt {2 + \sqrt 3 } }}{2}$ .
Now taking $\theta = {15^ \circ }$ and by substituting the values we have:
$\sin \dfrac{{{{15}^ \circ }}}{2} = \sqrt {\dfrac{{1 - \cos {{15}^ \circ }}}{2}}\; \Rightarrow \sin 7.5 = \sqrt {\dfrac{{1 - \dfrac{{\sqrt {2 + \sqrt 3 } }}{2}}}{2}}$ .
It gives us
$\sin 7.5 = \sqrt {\dfrac{{2 - \sqrt {2 + \sqrt 3 } }}{4}} \; \Rightarrow \sin 7.5 = \dfrac{{\sqrt {2 - \sqrt {2 + \sqrt 3 } } }}{2}$ .
Hence the exact value of $\sin 7.5$ is $\dfrac{{\sqrt {2 - \sqrt {2 + \sqrt 3 } } }}{2}$ .
So, the correct answer is “ $\dfrac{{\sqrt {2 - \sqrt {2 + \sqrt 3 } } }}{2}$ ”.

Note : We should keep in mind that in the half angle trigonometric formula, problems for sine or cosine we should always put the plus/minus sign in front of each square root. We should also check in which quadrant the value falls because our answer should be positive or negative, it depends on which quadrant the new half angle is. Here in the above question the given angle of sine falls in the first quadrant so the resulting answer is positive. It should be noted that the half angle formulas form double angle formulas.