Question

Write $\sin {35^{\text{o}}}$ in fraction form.

Answer 1

Here, we are going to use the identities of trigonometry and some rules of trigonometry to solve this problem. We use the formula $\sin (A + B) = \sin A\cos B + \sin B\cos A$ and also an approximation which is ${\text{if }}x{\text{ is a smaller value, then }}\sin x \approx x$ . Here, we use approximations also, for finding the value of irrational numbers.

Complete step-by-step solution:
Generally, sine of an angle is equal to the length of the side opposite to this angle divided by the length of the hypotenuse. And similarly, cosine value is equal to, length of adjacent side divided by length of hypotenuse.
The values of angles are taken in radians and in degrees and ${1^0} = \pi {\text{ radians}}$ .
We all know that, ${30^0} = \dfrac{\pi }{6}$ and the value of $\sin \dfrac{\pi }{6}$ which is equal to $\dfrac{1}{2}$ . And $\cos \dfrac{\pi }{6} = \dfrac{{\sqrt 3 }}{2}$ .
And ${35^0} = {30^0} + {5^0}$
So, by using the formula $\sin (A + B) = \sin A\cos B + \sin B\cos A$ ,
We get, $\sin {35^0} = \sin ({30^0} + {5^0})$
$\Rightarrow \sin {35^0} = \sin {30^0}\cos {5^0} + \sin {5^0}\cos {30^0}$
As ${5^0}$ is a smaller angle, we take $\sin {5^0} \approx {5^0} = \dfrac{\pi }{{36}}$
And as ${5^0}$ is a smaller angle, we also take, $\cos {5^0} \approx 1$
$\Rightarrow \sin {35^0} = \left( {\dfrac{1}{2}} \right)\left( 1 \right) + \left( {\dfrac{\pi }{{36}}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)$
The value of $\sqrt 3$ is approximately equal to $1.732$ . So, $\dfrac{{\sqrt 3 }}{2}$ has a value approximately equal to $0.866$.
So, here $\dfrac{{\sqrt 3 }}{2}$ is approximately equal to $\dfrac{{43}}{{50}}$ .
$\Rightarrow \sin {35^0} = \left( {\dfrac{1}{2}} \right)\left( 1 \right) + \left( {\dfrac{\pi }{{36}}} \right)\left( {\dfrac{{43}}{{50}}} \right)$
$\Rightarrow \sin {35^0} = \left( {\dfrac{1}{2}} \right)\left( 1 \right) + \left( {\dfrac{{22}}{7}} \right)\left( {\dfrac{1}{{36}}} \right)\left( {\dfrac{{43}}{{50}}} \right)$ (the vale of $\pi$ is equal to $\dfrac{{22}}{7}$ )
On simplification, we get,
$\Rightarrow \sin {35^0} = \left( {\dfrac{1}{2}} \right) + \left( {\dfrac{{473}}{{6300}}} \right)$
$\Rightarrow \sin {35^0} = \dfrac{{3623}}{{6300}}$
In decimal form, its value is approximately equal to $0.5750$ .
This is the required fraction form.

Note: We took approximations which have three digits after the decimal point on evaluation. So, we got a huge fraction here. As accurate the approximation will be, that much accurately, we get our answer. Also, there is another formula which is $\cos (A + B) = \cos A\cos B - \sin A\sin B$ .
The approximations are always valid only if the angle is very less. Otherwise, the approximation is not valid and it shouldn’t be taken.
When compared to thirty, five is a lesser value. So, the approximation is valid.