Question

find the cos 67.5° 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 cos 67.5° = √? - √ /

Explanation:

Step1: Identify the half - angle formula and θ

We know that the half - angle formula for cosine is \(\cos(\frac{\theta}{2})=\pm\sqrt{\frac{1 + \cos\theta}{2}}\). We want to find \(\cos(67.5^{\circ})\), and since \(67.5^{\circ}=\frac{135^{\circ}}{2}\), then \(\theta = 135^{\circ}\). Also, \(67.5^{\circ}\) is in the first quadrant where cosine is positive, so we take the positive sign. So \(\cos(67.5^{\circ})=\cos(\frac{135^{\circ}}{2})=\sqrt{\frac{1+\cos(135^{\circ})}{2}}\).

Step2: Find the value of \(\cos(135^{\circ})\)

We know that \(\cos(135^{\circ})=\cos(180^{\circ}- 45^{\circ})=-\cos(45^{\circ})=-\frac{\sqrt{2}}{2}\).

Step3: Substitute \(\cos(135^{\circ})\) into the formula

Substitute \(\cos(135^{\circ})=-\frac{\sqrt{2}}{2}\) into \(\sqrt{\frac{1+\cos(135^{\circ})}{2}}\):

$$
LATEXBLOCK0
$$

Answer:

\(\cos 67.5^{\circ}=\frac{\sqrt{2-\sqrt{2}}}{2}\) (So the first box is \(2\), the second box is \(2\), and the denominator box is \(2\))