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To find the sine and cosine of \(22.5^\circ\), we can use the half-angle identities for sine and cosine. We know that \(22.5^\circ\) is half of \(45^\circ\), and the half-angle identities are given by:
[tex]\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos\theta}{2}} \][/tex]
[tex]\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos\theta}{2}} \][/tex]
Using these identities with \(\theta = 45^\circ\):
1. Calculate \(\cos(45^\circ)\):
[tex]\[ \cos(45^\circ) = \frac{\sqrt{2}}{2} \][/tex]
2. Find \(\sin(22.5^\circ)\):
[tex]\[ \sin\left(22.5^\circ\right) = \sin\left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 - \cos(45^\circ)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \][/tex]
Simplify inside the square root:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{2/2 - \sqrt{2}/2}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
Therefore:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
3. Find \(\cos(22.5^\circ)\):
[tex]\[ \cos\left(22.5^\circ\right) = \cos\left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 + \cos(45^\circ)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \][/tex]
Simplify inside the square root:
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{2/2 + \sqrt{2}/2}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} = \sqrt{\frac{\sqrt{2}}{2} + \frac{1}{2}} \][/tex]
Therefore:
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \][/tex]
Now, the exact values of the sine and cosine of \(22.5^\circ\) are:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \][/tex]
Next, we provide the approximations of these values to three decimal places using decimal notation:
[tex]\[ \sin\left(22.5^\circ\right) \approx 0.383 \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) \approx 0.924 \][/tex]
So, the step-by-step solution is summarized as follows:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \][/tex]
Approximations to three decimal places:
[tex]\[ \sin\left(22.5^\circ\right) \approx 0.383 \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) \approx 0.924 \][/tex]
[tex]\[ \sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos\theta}{2}} \][/tex]
[tex]\[ \cos\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \cos\theta}{2}} \][/tex]
Using these identities with \(\theta = 45^\circ\):
1. Calculate \(\cos(45^\circ)\):
[tex]\[ \cos(45^\circ) = \frac{\sqrt{2}}{2} \][/tex]
2. Find \(\sin(22.5^\circ)\):
[tex]\[ \sin\left(22.5^\circ\right) = \sin\left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 - \cos(45^\circ)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} \][/tex]
Simplify inside the square root:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{2/2 - \sqrt{2}/2}{2}} = \sqrt{\frac{2 - \sqrt{2}}{4}} = \sqrt{\frac{2 - \sqrt{2}}{2 \cdot 2}} = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
Therefore:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
3. Find \(\cos(22.5^\circ)\):
[tex]\[ \cos\left(22.5^\circ\right) = \cos\left(\frac{45^\circ}{2}\right) = \sqrt{\frac{1 + \cos(45^\circ)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} \][/tex]
Simplify inside the square root:
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{2/2 + \sqrt{2}/2}{2}} = \sqrt{\frac{2 + \sqrt{2}}{4}} = \sqrt{\frac{2 + \sqrt{2}}{2 \cdot 2}} = \sqrt{\frac{\sqrt{2}}{2} + \frac{1}{2}} \][/tex]
Therefore:
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \][/tex]
Now, the exact values of the sine and cosine of \(22.5^\circ\) are:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \][/tex]
Next, we provide the approximations of these values to three decimal places using decimal notation:
[tex]\[ \sin\left(22.5^\circ\right) \approx 0.383 \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) \approx 0.924 \][/tex]
So, the step-by-step solution is summarized as follows:
[tex]\[ \sin\left(22.5^\circ\right) = \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) = \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \][/tex]
Approximations to three decimal places:
[tex]\[ \sin\left(22.5^\circ\right) \approx 0.383 \][/tex]
[tex]\[ \cos\left(22.5^\circ\right) \approx 0.924 \][/tex]
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