Question

find the sin 112.5° using the half-angle formula. rationalize the denominator if necessary. half-angle formulas sin(θ/2) = ±√(1 - cos(θ))/2 cos(θ/2) = ±√(1 + cos(θ))/2 sin 112.5° = √? + √□/□

Explanation:

Step1: Determine θ for half - angle

We know that \(112.5^{\circ}=\frac{\theta}{2}\), so we can solve for \(\theta\) by multiplying both sides by 2. We get \(\theta = 2\times112.5^{\circ}=225^{\circ}\).

Step2: Apply the half - angle formula for sine

The half - angle formula for sine is \(\sin(\frac{\theta}{2})=\pm\sqrt{\frac{1 - \cos\theta}{2}}\). Since \(112.5^{\circ}\) is in the second quadrant where sine is positive, we use the positive sign. So \(\sin(112.5^{\circ})=\sin(\frac{225^{\circ}}{2})=\sqrt{\frac{1-\cos(225^{\circ})}{2}}\).

Step3: Find the value of \(\cos(225^{\circ})\)

We know that \(225^{\circ}=180^{\circ} + 45^{\circ}\), and using the cosine of a sum formula \(\cos(A + B)=\cos A\cos B-\sin A\sin B\), or we can recall that \(\cos(225^{\circ})=-\frac{\sqrt{2}}{2}\).

Step4: Substitute \(\cos(225^{\circ})\) into the formula

Substitute \(\cos(225^{\circ})=-\frac{\sqrt{2}}{2}\) into \(\sqrt{\frac{1-\cos(225^{\circ})}{2}}\):

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Answer:

\(\sin112.5^{\circ}=\frac{\sqrt{2+\sqrt{2}}}{2}\) (So the values in the boxes are 2, 2, and 2 respectively)