Graphing All Six Trig Functions

Pre-Calculus · Axiom Academy

EXAMPLE Graphing All Six Trig Functions Read each curve's period, range, and asymptotes from its graph — and see why the reciprocal pairs line up. Identify the key features of all six trigonometric functions — , , , , , — from their graphs: the period , the range , and where the vertical asymptotes fall. We work through them in three reciprocal-linked pairs. Both are smooth waves with no breaks. cos x is just sin x shifted left by π/2. tan climbs upward between its asymptotes; cot falls downward between its own. Each repeats every π. Each U-branch hugs an asymptote on both sides and touches its faint parent wave exactly at the parent's max or min (where |value| = 1). The waves (sin, cos) stay inside [−1, 1]; tan and cot sweep the whole vertical range; sec and csc live outside [−1, 1] in U-branches. Nice work — you read the period, range, and asymptotes of all six trig functions straight off their graphs. Waves (sin, cos): period , range [-1, 1] , no asymptotes — continuous everywhere. tan, cot: period (half the waves'), range all reals; tan has asymptotes at , cot at . sec, csc: period , range , U-branches that touch their parent wave at its extrema; sec breaks where , csc where . Shared breaks: and share asymptotes (both ); and share asymptotes (both ). Once you know which trig ratio sits in the denominator, the asymptotes — and the whole shape — follow for free.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.