Question

find the exact value without a calculator. double - angle formulas: sin(2θ)=2 sinθ cosθ cos(2θ)=cos²θ - sin²θ tan(2θ)=2 tanθ / (1 - tan²θ) half - angle formulas: sin(θ / 2)=±√((1 - cosθ)/2) cos(θ / 2)=±√((1 + cosθ)/2) tan(θ / 2)=±√((1 - cosθ)/(1 + cosθ)) sin15°=√(?-√ )/

Explanation:

Step1: Use half - angle formula for sine

Since \(15^{\circ}=\frac{30^{\circ}}{2}\), and \(\sin\frac{\theta}{2}=\pm\sqrt{\frac{1 - \cos\theta}{2}}\). For \(\theta = 30^{\circ}\), \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\).

$$ \sin15^{\circ}=\sin\frac{30^{\circ}}{2}=\sqrt{\frac{1-\cos30^{\circ}}{2}}=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}=\sqrt{\frac{2 - \sqrt{3}}{4}}=\frac{\sqrt{2-\sqrt{3}}}{2} $$

Answer:

\(\sin15^{\circ}=\frac{\sqrt{2 - \sqrt{3}}}{2}\), so the numerator under the first square - root is \(2\), the number under the second square - root is \(3\), and the denominator is \(2\).