Question

If Sec (4A) = Cosec (A-${\text{2}}{{\text{0}}^0}$), where 4A is an acute angle, find the value of A.

Answer 1

Hint - Use the trigonometric ratios and covert the sec into cosec to simplify the given equation and solve for A. Sec and Cosec are complementary angles.

Complete step by step answer:
From Trigonometric ratios, we know that
Cosec (90 - x) = Sec (x) -- Equation (1) ----- Complementary Angles
Given,
Sec (4A) = Cosec (A - 20)
We can write, Sec (4A) = Cosec (90- 4A) Using Equation (1)
⟹Cosec (90-4A) = Cosec (A - 20)
The Cosec on both sides gets cancelled as its given 4A is an acute angle.
⟹90 - 4A = A – 20
⟹90 + 20 = A + 4A
⟹110 = 5A
⟹A =$\dfrac{{110}}{5} = {22^0}$
Therefore, The required angle A = 22 degrees.

Note – In such types of problems conversion of trigonometric ratios into a single ratio is a must as it simplifies the equation. Here sec and cosec are complementary angles, i.e., two angles are said to be complementary if their sum equals 90 degrees.