Sec, Cosec, Cot graph investigation

This investigation introduces the reciprocal trigonometric functions sec x cosec x and cot x through guided discovery rather than direct instruction. Students begin by sketching accurate graphs of sin x cos x and tan x, revisiting key features such as intercepts, turning points, periods and asymptotes. They then use these familiar graphs to explore the effect of taking reciprocals, making predictions about asymptotes, ranges and key points before constructing the graphs of the reciprocal functions themselves.

The activity encourages students to reason mathematically about the relationship between a function and its reciprocal, leading them to discover why secant and cosecant cannot take values between −1 and 1, how asymptotes arise, and how features such as period and symmetry are preserved. The lesson concludes with a reflection task that prompts students to generalise methods for sketching reciprocal trigonometric graphs and articulate the key connections between the original and reciprocal functions