use compound angles to solve $$ \cos 22^{\circ} $$

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To solve for $$ \cos 22^{\circ} $$ using compound angles, we can express $$ 22^{\circ} $$ as a combination of angles whose cosine values we know. A common approach is to use the angle sum or difference identities.

We can express $$ 22^{\circ} $$ as:

$$
22^{\circ} = 45^{\circ} - 23^{\circ}
$$

Using the cosine of a difference identity:

$$
\cos(a - b) = \cos a \cos b + \sin a \sin b
$$

In our case, $$ a = 45^{\circ} $$ and $$ b = 23^{\circ} $$. Thus, we have:

$$
\cos 22^{\circ} = \cos(45^{\circ} - 23^{\circ}) = \cos 45^{\circ} \cos 23^{\circ} + \sin 45^{\circ} \sin 23^{\circ}
$$

Now, we know:

$$
\cos 45^{\circ} = \frac{\sqrt{2}}{2}, \quad \sin 45^{\circ} = \frac{\sqrt{2}}{2}
$$

So we can substitute these values into the equation:

$$
\cos 22^{\circ} = \frac{\sqrt{2}}{2} \cos 23^{\circ} + \frac{\sqrt{2}}{2} \sin 23^{\circ}
$$

This simplifies to:

$$
\cos 22^{\circ} = \frac{\sqrt{2}}{2} (\cos 23^{\circ} + \sin 23^{\circ})
$$

Next, we need to find $$ \cos 23^{\circ} $$ and $$ \sin 23^{\circ} $$. We can use a calculator or trigonometric tables to find these values.

Let's calculate $$ \cos 23^{\circ} $$ and $$ \sin 23^{\circ} $$ and then substitute them back into the equation.

Calculating these values:

$$
\cos 23^{\circ} \approx 0.9205, \quad \sin 23^{\circ} \approx 0.3907
$$

Now substituting these values back into the equation:

$$
\cos 22^{\circ} = \frac{\sqrt{2}}{2} (0.9205 + 0.3907)
$$

Calculating the sum:

$$
0.9205 + 0.3907 \approx 1.3112
$$

Now, multiplying by $$ \frac{\sqrt{2}}{2} $$:

$$
\cos 22^{\circ} \approx \frac{\sqrt{2}}{2} \times 1.3112
$$

Calculating this gives:

$$
\cos 22^{\circ} \approx 0.926
$$

Thus, the final result is:

$$
\cos 22^{\circ} \approx 0.926
$$
$$ \cos 22^{\circ} \approx 0.926 $$

use compound angles to solve \( \cos 22^{\circ} \)

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