Question

If \( \sin(20^\circ + \theta) = \cos 30^\circ \), then the value of \( \theta \) is:

• A. \( 30^\circ \)
• B. \( 40^\circ \)
• C. \( 50^\circ \)
• D. \( 60^\circ \)

Hint:

Using the identity: \[ \sin x = \cos (90^\circ - x) \] convert the equation into sine terms and solve for \( \theta \).

Answer 1

The Correct Option is C

Using the trigonometric identity:
\[ \sin x = \cos(90^\circ - x) \] We rewrite the given equation:
\[ \sin(20^\circ + \theta) = \cos 30^\circ \] \[ \sin(20^\circ + \theta) = \sin(90^\circ - 30^\circ) \] \[ \sin(20^\circ + \theta) = \sin 60^\circ \] Since the sine values are equal, we equate the angles:
\[ 20^\circ + \theta = 60^\circ \] \[ \theta = 60^\circ - 20^\circ \] \[ \theta = 50^\circ \]