Question

If \(\sin 3A = \cos (A - 26^{\circ})\), where 3A is an acute angle, find the value of A.

• A. \(29^{\circ}\)
• B. \(61^{\circ}\)
• C. \(51^{\circ}\)
• D. \(39^{\circ}\)

Hint:

For any equation of the form \(\sin X = \cos Y\) where both are acute angles, the angles must be complementary:
\[ X + Y = 90^{\circ} \]
Here, \(3A + (A - 26^{\circ}) = 90^{\circ} \implies 4A - 26^{\circ} = 90^{\circ} \implies 4A = 116^{\circ} \implies A = 29^{\circ}\).
This shortcut saves steps in matching co-functions.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This question is from Trigonometry, focusing on complementary trigonometric ratios.
We are given an equation relating the sine of one angle to the cosine of another, and we need to determine the value of \(A\).

Step 2: Key Formula or Approach:
We use the co-function trigonometric identity that relates sine and cosine:
\[ \sin \theta = \cos (90^{\circ} - \theta) \]
This allows us to convert the equation so that both sides have the same trigonometric function (cosine), which allows us to equate the angles directly.

Step 3: Detailed Explanation:
Given equation:
\[ \sin 3A = \cos (A - 26^{\circ}) \]
Using the identity \(\sin 3A = \cos (90^{\circ} - 3A)\), rewrite the left side:
\[ \cos (90^{\circ} - 3A) = \cos (A - 26^{\circ}) \]
Since both angles are acute, we can equate the arguments directly:
\[ 90^{\circ} - 3A = A - 26^{\circ} \]
Rearrange the equation to group the terms of \(A\) on one side and constants on the other:
\[ 90^{\circ} + 26^{\circ} = A + 3A \]
\[ 116^{\circ} = 4A \]
Divide both sides by 4 to solve for \(A\):
\[ A = \frac{116^{\circ}}{4} \]
\[ A = 29^{\circ} \]

Step 4: Final Answer:
The value of \(A\) is \(29^{\circ}\).