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

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: \[ \cos 22^{\circ} \approx 0.926 \]