Calculate in surd form, the value of \(\tan 15°\).
Question 1 Report
Calculate in surd form, the value of \(\tan 15°\).
Answer Details
To calculate the value of \(\tan 15°\), we can use the half-angle formula for tangent: \[\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta}\] Letting \(\theta = 30°\), we get: \[\tan 15° = \tan \frac{30°}{2} = \frac{\sin 30°}{1 + \cos 30°}\] We know that \(\sin 30° = \frac{1}{2}\) and \(\cos 30° = \frac{\sqrt{3}}{2}\). Substituting these values into the equation above, we get: \[\tan 15° = \frac{\frac{1}{2}}{1 + \frac{\sqrt{3}}{2}} = \frac{1}{2 + \sqrt{3}}\] Rationalizing the denominator, we get: \[\tan 15° = \frac{1}{2 + \sqrt{3}} \cdot \frac{2 - \sqrt{3}}{2 - \sqrt{3}} = \frac{2 - \sqrt{3}}{1} = \boxed{2 - \sqrt{3}}\] Therefore, the answer is (d) \(2 - \sqrt{3}\).
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