Trigonometric Expression Simplification: sin 30° / (1 + cos 30°)
The problem requires evaluating the following trigonometric expression:
$ \frac{\sin 30^\circ}{1 + \cos 30^\circ} + \frac{1 + \cos 30^\circ}{\sin 30^\circ} $
Algebraic Simplification Strategy
To solve this efficiently, we can simplify the expression algebraically before substituting the values.
Let the expression be $E$. Let $\theta = 30^\circ$. The expression becomes:
$ E = \frac{\sin \theta}{1 + \cos \theta} + \frac{1 + \cos \theta}{\sin \theta} $
Find a common denominator, which is $\sin \theta (1 + \cos \theta)$:
$ E = \frac{(\sin \theta)^2 + (1 + \cos \theta)^2}{\sin \theta (1 + \cos \theta)} $
Expand the numerator:
$ E = \frac{\sin^2 \theta + (1 + 2\cos \theta + \cos^2 \theta)}{\sin \theta (1 + \cos \theta)} $
Apply the Pythagorean identity $\sin^2 \theta + \cos^2 \theta = 1$:
$ E = \frac{(\sin^2 \theta + \cos^2 \theta) + 1 + 2\cos \theta}{\sin \theta (1 + \cos \theta)} $
$ E = \frac{1 + 1 + 2\cos \theta}{\sin \theta (1 + \cos \theta)} $
$ E = \frac{2 + 2\cos \theta}{\sin \theta (1 + \cos \theta)} $
Factor out 2 from the numerator:
$ E = \frac{2(1 + \cos \theta)}{\sin \theta (1 + \cos \theta)} $
Cancel the $(1 + \cos \theta)$ term (since $1 + \cos 30^\circ \neq 0$):
$ E = \frac{2}{\sin \theta} $
Substituting Values and Final Calculation
We know that $\sin 30^\circ = \frac{1}{2}$. Substitute this value into the simplified expression:
$ E = \frac{2}{\sin 30^\circ} $
$ E = \frac{2}{1/2} $
$ E = 2 \times 2 $
$ E = 4 $
The value of the expression is 4.
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