Question

How do you evaluate the sine, cosine and tangent of the angle 53 degrees without using the calculator?

Answer 1

Hint: We are given an angle of 53 degrees and we are asked to find the value of sin, cos and tan at 53 degrees without calculating. We will need to first understand how we can write sine and cosine in the polynomial form, then using the relation between sin and cosine we will get the value of tan. We also need to understand the radian and degree. We will be using \[{{180}^{\circ }}=\pi \text{ radian}\text{.}\]

Complete step by step answer:
We are given that we are having an angle in degree and we have to find the value of sin, cosine and tan at that angle. Now, to answer the question without a calculator we will first learn what the way of writing cosine and sine function is. Now, cosine function is a periodic function that is expressed as a polynomial function which is given as
\[\cos x=\sum\limits_{n=0}^{\infty }{\dfrac{{{\left( -1 \right)}^{n}}{{x}^{2n}}}{\left( 2n \right)!}}\]
Or we can expand it as
\[\cos x=1-\dfrac{{{x}^{2}}}{2!}+\dfrac{{{x}^{4}}}{4!}-\dfrac{{{x}^{6}}}{6!}.....\]
For sine function, its expansion as the polynomial function is given as
\[\sin x=\sum\limits_{n=0}^{\infty }{{{\left( -1 \right)}^{n}}\dfrac{{{x}^{2n+1}}}{\left( 2n+1 \right)!}}\]
Or we can expand it as
\[\sin x=x-\dfrac{{{x}^{3}}}{3!}+\dfrac{{{x}^{5}}}{5!}.....\]
where x is always in the radian.
Now, to find the value of sine, cosine and tan without a calculator we will use the polynomial expansion form to calculate it. Now we need the angle in radian. We have 53 degrees and we have to convert it into radian. Now, we know,
\[{{180}^{\circ }}=\pi \text{ radian}\]
So,
\[{{1}^{\circ }}=\dfrac{\pi }{180}\text{ radian}\]
Using the unitary method, we get
\[{{53}^{\circ }}=\dfrac{\pi }{180}\times \text{5}{{\text{3}}^{\circ }}\text{ radian}\]
On simplifying we get,
\[\Rightarrow {{53}^{\circ }}=0.925\text{ radian}\]
So, we have x = 0.925.
So, we use x = 0.925 in \[\cos x=1-\dfrac{{{x}^{2}}}{2!}+\dfrac{{{x}^{4}}}{4!}-\dfrac{{{x}^{6}}}{6!}.....\]
So, we get,
\[\cos \left( 0.925 \right)=1-\dfrac{{{\left( 0.925 \right)}^{2}}}{2!}+\dfrac{{{\left( 0.925 \right)}^{4}}}{4!}\]
On simplifying, we get,
\[\Rightarrow \cos \left( 0.925 \right)=0.6028\]
Also, we will put x = 0.925 in \[\sin x=x-\dfrac{{{x}^{3}}}{3!}+\dfrac{{{x}^{5}}}{5!}.....\]
So, we get,
\[\sin \left( 0.925 \right)=0.925-\dfrac{{{\left( 0.925 \right)}^{3}}}{3!}+\dfrac{{{\left( 0.925 \right)}^{5}}}{5!}\]
On simplifying, we get,
\[\Rightarrow \sin \left( 0.925 \right)=0.7986\]
Now as \[\tan \theta =\dfrac{\sin \theta }{\cos \theta }\] so we get,
\[\tan \left( 0.925 \right)=\dfrac{0.7986}{0.6018}\]
On solving, we get,
\[\Rightarrow \tan \left( 0.925 \right)=1.3270\]
Hence, we get the value of sin at 53 degrees as 0.7986 and value of cos at 53 degrees is 0.6018 and the value of tan at 53 degrees is 1.3270.

Note:
While using the polynomial form of sin or cosine, we need to be careful that in expansion, the variable is always in radian and so we need to convert them into radian first, if we do not change, then the value we get will be incorrect. Also, in expansion form we will use \[\dfrac{\sin \theta }{\cos \theta }\] to make things less complicated.