Find the exact value of cos 22.5 degree using the half-angle identity.
The Correct Answer and Explanation is :
To find the exact value of cos(22.5∘)\cos(22.5^\circ), we use the half-angle identity for cosine, which is: cos(θ2)=±1+cos(θ)2\cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos(\theta)}{2}}
Step-by-step solution:
- Identify the angle and the quadrant:
- Here, θ=45∘\theta = 45^\circ, and we are finding cos(θ2)=cos(22.5∘)\cos\left(\frac{\theta}{2}\right) = \cos(22.5^\circ).
- Since 22.5∘22.5^\circ is in the first quadrant, cos(22.5∘)>0\cos(22.5^\circ) > 0, so we take the positive square root.
- Use the half-angle identity: cos(22.5∘)=1+cos(45∘)2\cos(22.5^\circ) = \sqrt{\frac{1 + \cos(45^\circ)}{2}}
- Substitute cos(45∘)\cos(45^\circ): From trigonometric values, cos(45∘)=22\cos(45^\circ) = \frac{\sqrt{2}}{2}. Substituting this into the formula: cos(22.5∘)=1+222\cos(22.5^\circ) = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}}
- Simplify the expression: Combine terms inside the numerator: cos(22.5∘)=22+222\cos(22.5^\circ) = \sqrt{\frac{\frac{2}{2} + \frac{\sqrt{2}}{2}}{2}} cos(22.5∘)=2+222\cos(22.5^\circ) = \sqrt{\frac{\frac{2 + \sqrt{2}}{2}}{2}} cos(22.5∘)=2+24\cos(22.5^\circ) = \sqrt{\frac{2 + \sqrt{2}}{4}}
- Final simplification: cos(22.5∘)=2+22\cos(22.5^\circ) = \frac{\sqrt{2 + \sqrt{2}}}{2}
Explanation:
The key to solving this is the half-angle identity and knowing the exact value of cos(45∘)\cos(45^\circ). By substituting and carefully simplifying the fractions, we find that: cos(22.5∘)=2+22\cos(22.5^\circ) = \frac{\sqrt{2 + \sqrt{2}}}{2}
This exact form highlights the importance of simplifying nested fractions and square roots systematically.