Track tangent and reciprocal-function domains

Try

At a unit-circle point, divide vertical coordinate by horizontal coordinate. This ratio is tan θ=sin θ/cos θ. At θ=π/4 it is 1. At θ=3π/4 it is −1. What happens when the horizontal coordinate is zero? The ratio is undefined; it is not a very large real value at that exact angle.

Predict

Predict whether adding π changes tangent. Both circle coordinates change sign, leaving their ratio unchanged. Tangent therefore has period π even though sine and cosine each have period 2π. Predict whether its zeros occur where sine or cosine is zero.

Experiment

Sketch one tangent branch on −π/2<θ<π/2. Mark its zero at 0 and points (−π/4,−1), (π/4,1). Draw dashed boundary lines at ±π/2, with no filled point on the curve there.

Function	Definition	Denominator must not vanish
tan θ	sin θ / cos θ	cos θ
sec θ	1 / cos θ	cos θ
csc θ	1 / sin θ	sin θ
cot θ	cos θ / sin θ	sin θ

Use exact circle values to add one valid value and one excluded angle for each row. Notice that secant and cosecant cannot be zero: their numerators are 1.

Understand

For any integer k, cosine vanishes at π/2+kπ and sine vanishes at kπ. These describe all exclusions in the table. Cotangent has period π; secant and cosecant have period 2π. Since sine and cosine have magnitudes at most 1, the defined secant and cosecant values have magnitude at least 1. Their graphs lie at or above 1 or at or below −1.

Near π/2, sine stays positive. Cosine is positive on the left and negative on the right, with magnitude tending to zero. Consequently tangent grows without bound positively from the left and negatively from the right. Sign reasoning explains the two branches; a graphing application must not connect them through the asymptote.

The expression 1/tan θ is not interchangeable with cot θ at every input. Cotangent is defined at π/2 and equals zero, but tangent is undefined there, so its reciprocal cannot be evaluated. The expressions agree on their shared domain, where both sine and cosine are nonzero. A simplified identity must retain original exclusions.

Apply

For a requested value, write its sine/cosine ratio first. For a graph claim, list zeros, excluded angles and period separately. In the written identity task, give a point at which domains differ rather than treating equal formulas as unrestricted functions.

Continue

Inverse trigonometric functions are different from reciprocals. They return selected angles, and the next lesson makes their principal ranges explicit.