Question

If $\sec 4A = \cos ec(A - {20^ \circ }),$ where $4A$ is an acute angle, find the value of $A$ .

Answer 1

This is related to trigonometric ratios of complementary angles; two angles are said to be complementary if their sums equals $90$ degrees. So just convert it to the same trigonometric functions, hence we can substitute their angles as the same.

Complete step-by-step answer:
As we know that $\cos ec(90 - x) = \sec x$
From the left hand side we get $\sec 4A$ as $\cos ec(90 - 4A)$ , hence substitute $\sec 4A$ with $\cos ec(90 - 4A)$ .
$\Rightarrow \cos ec(90 - 4A) = \cos ec(A - {20^ \circ })$
As both left-hand side and right-hand side is in terms of $\cos ec$ , hence we can equate their angles as equal.
$\Rightarrow 90 - 4A = A - 20$
By rearranging the terms, we get,
$\Rightarrow 90 + 20 = 4A + A$
Perform the arithmetic operation,
$\Rightarrow 110 = 5A$
Again, rearranging the terms, we get,
$\Rightarrow 5A = 110$
$\Rightarrow A = \dfrac{{110}}{5}$
Hence $A = 22$
Therefore,
The value of $A$ is $A = 22$ degrees.

Additional Information: The trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. ... The most widely used trigonometric functions are the sine, the cosine, and the tangent.
The complementary angles are the set of two angles such that their sum is equal to ${90^ \circ }$ . For example, ${30^ \circ }$ and ${60^ \circ }$ are complementary to each other as their sum is equal to ${90^ \circ }$ .
$\sin$ of an angle $=$ $\cos$ of its complementary angle.
Thus, the measure of the acute angle $A$ can be easily calculated by making use of trigonometry ratio of complementary angles.

Note: We have to check whether the given question is related to trigonometric ratios of complementary angles, then their angles must be in complementary which means that we can write their angles sum equals $90$ degrees.