Tangent: ⁦graph, domain and period⁩

⁦This studio is for Iranian grade 12 (trigonometry⁩: ⁦periodicity and tangent⁩). ⁦The x-axis counts in degrees (one square is 90 degrees) and 180 degrees is π radians⁩. ⁦The y-axis counts in ones⁩.

⁦On the⁩ الدائرة المثلثية, ⁦the line touching the circle at A, the point (١,٠), is the tangent axis⁩. ⁦Extend the second arm of the angle α until it meets this axis; the signed length from A to the meeting point is tanα⁩. ⁦It is positive in quadrants 1 and 3 and negative in quadrants 2 and 4⁩.

⁦As α nears 90 degrees the tangent grows larger and larger⁩. ⁦At 90 and 270 degrees the second arm is parallel to the tangent axis and never meets it, so tan ٩٠° is undefined and the graph is made of separate branches⁩.

⁦The tangent repeats every 180 degrees (π), half the period of the sine⁩. ⁦Since tanα equals sinαcosα, it is undefined wherever the cosine is zero⁩.

⁦Common mistakes⁩: ⁦taking the tangent as positive in quadrant 2; putting the breaks at the zeros of the tangent; a period of 360 degrees as for the sine; and finding the second solution as for the sine with ١٨٠° − x⁩.