Question

Find $\tan {53^ \circ }38'$
A. 1.23
B. 0.45
C. 0.56
D. 1.35

Answer 1

Hint: We know that the degree is a measure of an angle, it is also known as arc degree because a plane angle is a measurement of full rotation of 360°. Since degree is a frequently used unit to measure an angle one may confuse it to be an S.I unit but the S.I unit for angle measurement is radian.

Complete step-by-step answer:
Since one full rotation makes an angle 2π. So one degree is $ \dfrac{{2\pi }}{{360}} = \dfrac{\pi }{{180}} $ radians. A degree is further subdivided in minutes and seconds. This notation is DMS notation represented by degree-minutes –seconds. One degree is divided into
 $\Rightarrow {1^ \circ } = 60' $ minutes in an arc
 $\Rightarrow 1' = 60'' $ (one minute is equal to 60 seconds)
Now let us convert the above angle into a complete degree for this we will have to convert 38’ into degrees and add it to 53°,
 $
\Rightarrow 60' = {1^ \circ } \\
\Rightarrow 1' = {\left( {\dfrac{1}{{60}}} \right)^ \circ } \\
\Rightarrow 38' = {\left( {\dfrac{{38}}{{60}}} \right)^ \circ } = {0.63^ \circ } \\
  $
Therefore, now the angle gets converted and the final value to be found out is
 $\Rightarrow \tan {53^ \circ }38' = \tan {53.63^ \circ } $
We know that tangent of A can be written as
 $\Rightarrow \tan A = \dfrac{{\sin A}}{{\cos A}} $
Similarly
 $
\Rightarrow \tan {53^ \circ }38' = \tan {53.63^ \circ } \\
\Rightarrow \tan {53.63^ \circ } = \dfrac{{\sin {{53.63}^ \circ }}}{{\cos {{53.63}^ \circ }}} = 1.35 \\
  $
We can now calculate the value for this angle using a calculator or from a trigonometric value table, because we don’t have formula or trigonometric properties to calculate the value for this angle like we do for other trigonometric angles like 45°, 15°etc. therefore we have to remember this value as it is.

Note: All the trigonometric functions have got a very important property in common that is periodicity. Remember that the trigonometric ratios are real numbers as long as angle A is real. Trigonometric functions are also called circular functions.