Solving Trigonometric Equation: sin x = cos(5x - 60°)

We are given the equation: $ \sin x = \cos(5x - 60^\circ) $ To solve this, we use the trigonometric identity $ \sin \theta = \cos(90^\circ - \theta) $. Applying this identity, we get:

General Solution for Cosine Equality

If $ \cos A = \cos B $, then the general solution is $ A = \pm B + n \cdot 360^\circ $, where $ n $ is an integer. Applying this to our equation:

Case 1: Positive sign

$ 90^\circ - x = 5x - 60^\circ + n \cdot 360^\circ $

Rearranging the terms to solve for $ x $: $ 90^\circ + 60^\circ = 5x + x + n \cdot 360^\circ $ $ 150^\circ = 6x + n \cdot 360^\circ $ $ 6x = 150^\circ - n \cdot 360^\circ $ $ x = \frac{150^\circ}{6} - \frac{n \cdot 360^\circ}{6} $ $ x = 25^\circ - n \cdot 60^\circ $

Case 2: Negative sign

$ 90^\circ - x = -(5x - 60^\circ) + n \cdot 360^\circ $

$ 90^\circ - x = -5x + 60^\circ + n \cdot 360^\circ $

Rearranging the terms: $ -x + 5x = 60^\circ - 90^\circ + n \cdot 360^\circ $ $ 4x = -30^\circ + n \cdot 360^\circ $ $ x = \frac{-30^\circ}{4} + \frac{n \cdot 360^\circ}{4} $ $ x = -7.5^\circ + n \cdot 90^\circ $

Finding the Value of x

We need to find a value of $ x $ that matches one of the options. Let's test values of $ n $ starting from $ n=0 $ in both cases.

• From Case 1 ($ x = 25^\circ - n \cdot 60^\circ $):
  • If $ n=0 $, $ x = 25^\circ - 0 \cdot 60^\circ = 25^\circ $. This matches option B.
  • If $ n=1 $, $ x = 25^\circ - 1 \cdot 60^\circ = -35^\circ $.
  • If $ n=-1 $, $ x = 25^\circ - (-1) \cdot 60^\circ = 25^\circ + 60^\circ = 85^\circ $.
• From Case 2 ($ x = -7.5^\circ + n \cdot 90^\circ $):
  • If $ n=0 $, $ x = -7.5^\circ $.
  • If $ n=1 $, $ x = -7.5^\circ + 90^\circ = 82.5^\circ $.

The value $ x = 25^\circ $ satisfies the equation and is present in the options.

Verification

Substitute $ x = 25^\circ $ into the original equation:

Left side: $ \sin(25^\circ) $

Right side: $ \cos(5 \cdot 25^\circ - 60^\circ) = \cos(125^\circ - 60^\circ) = \cos(65^\circ) $

Since $ \sin(25^\circ) = \cos(90^\circ - 25^\circ) = \cos(65^\circ) $, the equation holds true.

Thus, the value of $ x $ is $ 25^\circ $.