Question

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

Hint:

When you see an equation with sine on one side and cosine on the other, like \(\sin(X) = \cos(Y)\), it implies that the angles are complementary, i.e., \(X + Y = 90^\circ\). In this case, \(3A + (A - 26^\circ) = 90^\circ\), which gives \(4A = 116^\circ\) and \(A=29^\circ\). This is a quick shortcut.

Answer 1

Step 1: Understanding the Concept:
To solve this trigonometric equation, we need to use the complementary angle identities to express both sides of the equation with the same trigonometric function.

Step 2: Key Formula or Approach:
We will use the identity \(\sin\theta = \cos(90^\circ - \theta)\). By applying this, we can make both sides of the equation a cosine function and then equate the angles.

Step 3: Detailed Explanation:
The given equation is: \[ \sin 3A = \cos(A - 26^\circ) \] Using the identity, we can rewrite the LHS as: \[ \sin 3A = \cos(90^\circ - 3A) \] Now substitute this into the equation: \[ \cos(90^\circ - 3A) = \cos(A - 26^\circ) \] Since both 3A and (A - 26\(^\circ\)) must represent angles for which the cosine function is defined and equal, we can equate the angles (assuming they are in the primary range): \[ 90^\circ - 3A = A - 26^\circ \] Now, solve for A by rearranging the terms: \[ 90^\circ + 26^\circ = A + 3A \] \[ 116^\circ = 4A \] \[ A = \frac{116^\circ}{4} \] \[ A = 29^\circ \] We should check if the condition that 3A is an acute angle is met. \(3A = 3 \times 29^\circ = 87^\circ\). Since \(87^\circ < 90^\circ\), 3A is indeed an acute angle.

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