3) A pilot flying from Kingsport to Chattanooga flies 22^circ off course for 40 miles before discovering his error. If the direct air distance between the two cities is 210 miles. ) how far is the pilot from Chattanooga when he finds his mistake?

## Step 1<br />The problem involves the use of trigonometry, specifically the cosine function. The pilot's deviation from the direct route forms a right triangle, where the hypotenuse is the direct air distance between the two cities (210 miles), and the angle of deviation is \(22^{\circ}\).<br /><br />## Step 2<br />The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse. In this case, the adjacent side is the distance the pilot has flown off course (40 miles), and the hypotenuse is the direct air distance between the two cities (210 miles).<br /><br />### **The formula for cosine is: \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)**<br /><br />## Step 3<br />We can rearrange the formula to solve for the adjacent side:<br /><br />### **The formula to find the adjacent side is: \(\text{adjacent side} = \cos(\theta) \times \text{hypotenuse}\)**<br /><br />## Step 4<br />Substitute the given values into the formula:<br /><br />### **The formula becomes: \(\text{adjacent side} = \cos(22^{\circ}) \times 210\)**<br /><br />## Step 5<br />Calculate the value of \(\cos(22^{\circ})\) and multiply it by 210.<br /><br />## Step 6<br />Subtract the calculated value from the direct air distance (210 miles) to find the distance the pilot is from Chattanooga when he finds his mistake.