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Question: Find \(\tan {53^ \circ }38'\) A. 1.23 B. 0.45 C. 0.56 D. 1.35...

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

Explanation

Solution

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 2π360=π180\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
⇒1∘=60′\Rightarrow {1^ \circ } = 60' minutes in an arc
⇒1′=60′′\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°,
⇒60′=1∘ ⇒1′=(160)∘ ⇒38′=(3860)∘=0.63∘  \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
⇒tan⁡53∘38′=tan⁡53.63∘\Rightarrow \tan {53^ \circ }38' = \tan {53.63^ \circ }
We know that tangent of A can be written as
⇒tan⁡A=sin⁡Acos⁡A\Rightarrow \tan A = \dfrac{{\sin A}}{{\cos A}}
Similarly
⇒tan⁡53∘38′=tan⁡53.63∘ ⇒tan⁡53.63∘=sin⁡53.63∘cos⁡53.63∘=1.35  \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.