Trigonometric Equation Setup
The problem asks us to find the value of angle A from the given trigonometric equation: \( \tan (3A) = \cot (A - 22^\circ) \) We are also given a crucial condition: \(3A\) must be an acute angle. An acute angle is an angle greater than \(0^\circ\) and less than \(90^\circ\) (i.e., \(0^\circ < 3A < 90^\circ\)).
Tan-Cot Identity for Complementary Angles
To solve this equation, we can use a standard trigonometric identity that relates the tangent and cotangent functions. The identity is:
\( \cot(\theta) = \tan(90^\circ - \theta) \)
Let's apply this identity to the \(\cot (A - 22^\circ)\) part of our equation. Here, \(\theta = A - 22^\circ\). Substituting this into the identity, we get:
\( \cot (A - 22^\circ) = \tan\left(90^\circ - (A - 22^\circ)\right) \)
Now, simplify the expression inside the tangent function:
\( 90^\circ - (A - 22^\circ) = 90^\circ - A + 22^\circ = 112^\circ - A \)
So, we can rewrite the original equation by substituting this back:
\( \tan (3A) = \tan (112^\circ - A) \)
Calculating the Value of Angle A
When the tangents of two angles are equal, \(\tan(x) = \tan(y)\), it means the angles are either equal or differ by a multiple of \(180^\circ\) (\(x = y + n \cdot 180^\circ\), where \(n\) is an integer). Given the context of acute angles, we often consider the simplest case where the angles are related.
From \(\tan (3A) = \tan (112^\circ - A)\), we can set the arguments equal:
\( 3A = 112^\circ - A \)
Now, we solve this linear equation for A:
- Move the term involving A to the left side by adding A to both sides: \(3A + A = 112^\circ\)
- Combine the terms on the left side: \(4A = 112^\circ\)
- Isolate A by dividing both sides by 4: \(A = \frac{112^\circ}{4} \)
- Perform the division: \(A = 28^\circ\)
Checking the Acute Angle Constraint
We must verify if the value \(A = 28^\circ\) satisfies the condition that \(3A\) is an acute angle (\(0^\circ < 3A < 90^\circ\)).
Let's calculate \(3A\) using our result:
\( 3A = 3 \times 28^\circ = 84^\circ \)
Since \(84^\circ\) is between \(0^\circ\) and \(90^\circ\), the condition is met. \(3A\) is indeed an acute angle.
We can also check if the original equation holds true for \(A = 28^\circ\):
- Left Hand Side (LHS): \(\tan(3A) = \tan(84^\circ)\)
- Right Hand Side (RHS): \(\cot(A - 22^\circ) = \cot(28^\circ - 22^\circ) = \cot(6^\circ)\)
Using the complementary angle identity \(\tan(\alpha) = \cot(90^\circ - \alpha)\), we see that \(\tan(84^\circ) = \cot(90^\circ - 84^\circ) = \cot(6^\circ)\). Therefore, LHS = RHS, and the solution is correct.
Determining the Final Value of A
Based on the calculations and verification, the value of angle A that satisfies the equation \(\tan (3A) = \cot (A - 22^\circ)\) with \(3A\) being an acute angle is \(28^\circ\).
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