A wire meets a pole 47 feet up and a second wire meets it 40 feet up, both anchored 50 feet from the pole. What angle separates the wires?

1. Only the two legs of each right triangle are known, so tangent is the function to use. 2. Let b be the angle of the higher wire, so tan b = 47/50, and let a be the angle between the wires, so tan(b - a) = 40/50 = 4/5. 3. Substitute into tan(b - a) = (tan b - tan a) / (1 + tan b tan a) to get 4/5 = (47/50 - tan a) / (1 + (47/50) tan a). 4. Cross multiplying gives 4(1 + 0.94 tan a) = 5(0.94 - tan a), that is 4 + 3.76 tan a = 4.7 - 5 tan a. 5. Collecting terms gives 8.76 tan a = 0.7, so tan a is about 0.07991. 6. Taking the inverse tangent gives about 0.079739 radians. 7. Converting with the factor 180 / pi gives about 4.57 degrees. 8. The figure 8.76 is only the coefficient from the collected equation, not an angle, so it is a trap for stopping one line early. _Source: OpenStax Algebra and Trigonometry (CC BY 4.0), Ch 9 "Trigonometric Identities and Equations", section 9.2 Sum and Difference Identities_