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Question: Express \(\sec \;{50^ \circ } + \cot \;{78^ \circ }\) in terms of t-ratios of angles between \({0^ \...

Express sec⁡  50∘+cot⁡  78∘\sec \;{50^ \circ } + \cot \;{78^ \circ } in terms of t-ratios of angles between 0∘{0^ \circ } and 45∘{45^ \circ }

Explanation

Solution

The given question is the trigonometric expression and in order to express it in other angels we have to use the properties of trigonometric functions. We also need to know about complementary angles and ratios of complementary angles. Complementary angles are the angles which add up to90∘{90^ \circ }. In order to solve this question, we’ll figure out the angles at which conversions take place.

Formula used:
sec⁡(90−θ)=cos⁡ecθ cot⁡(90−θ)=tan⁡θ  \sec (90 - \theta ) = \cos ec\theta \\\ \cot (90 - \theta ) = \tan \theta \\\

Complete step by step answer:
We are given,
sec⁡  50∘+cot⁡  78∘\sec \;{50^ \circ } + \cot \;{78^ \circ }
To convert, we’ll rewrite the angles as a complement of 90∘{90^ \circ }
⇒sec⁡  50∘+cot⁡  78∘=sec⁡(90−40)∘  +  cot⁡  (90−12)∘\Rightarrow \sec \;{50^ \circ } + \cot \;{78^ \circ } = \sec {(90 - 40)^ \circ }\; + \;\cot \;{(90 - 12)^ \circ }
Now we can replace the ratios with the ratios of their complementary angles.
⇒cosec⁡  40∘+tan⁡  12∘\Rightarrow co\sec \;{40^ \circ } + \tan \;{12^ \circ }
This is the required answer.

Note: To simplify the expressions containing trigonometry, we need to memorize the properties associated with it. Trigonometric Ratios portray the relationship between measurement of angles and the length of the side of a triangle. It will make questions easier to solve. It is suggested that while solving the question of trigonometry we should carefully scrutinize the pattern of the given function, relating it with identities and then we should apply the formulas according to the identity which has been observed. When we have trigonometric ratios with angles 90∘  and  270∘90{^ \circ }\;and\;{270^ \circ }or we can say all the angles which are odd multiples of 90∘{90^ \circ } in the form-
90+θ 90−θ 270+θ 270−θ  90 + \theta \\\ 90 - \theta \\\ 270 + \theta \\\ 270 - \theta \\\
The following conversions take place,
sin⁡θ↔cos⁡θ tan⁡θ↔cot⁡θ cos⁡ecθ↔sec⁡θ  \sin \theta \leftrightarrow \cos \theta \\\ \tan \theta \leftrightarrow \cot \theta \\\ \cos ec\theta \leftrightarrow \sec \theta \\\
Also, no conversion takes places when angles are even multiples of 90∘{90^ \circ }. There is a special case with 45∘{45^ \circ }, since the complement of 45∘{45^ \circ }is 45∘{45^ \circ }, so the trigonometric ratio will remain the same.