Tangent and Reciprocal Graphs

Develop tangent period and asymptotes, secant and cosecant.

Lesson 3 of 42 parts Practice Flashcards

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Tangent Period and Asymptotes

Objective: Explain tangent period and asymptotes, apply the rule, and justify relevant restrictions.

Tangent is sin x/cos x, so it is undefined at x=π/2+kπ. Its period is π because shifting both sine and cosine by π reverses both signs while preserving the quotient. For tan(Bx), B≠0, period is π/|B|. Between adjacent vertical asymptotes, tangent takes every real value. Its unbounded output means it has no amplitude.

Formulas

$Ttan=π/|B|; tan x=sin x/cos x$

Worked examples

Try itFor y=3 tan(7x), give the period as a coefficient of π.Show answer

Step 1: Tangent has base period π. Step 2: Divide π by |7|. Step 3: The period is 1/7π.

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Tangent Period and Asymptotes

Objective: Explain tangent period and asymptotes, apply the rule, and justify relevant restrictions.

Formulas

$Ttan=π/|B|; tan x=sin x/cos x$

Worked examples

Try itFor y=3 tan(7x), give the period as a coefficient of π.Show answer

Step 1: Tangent has base period π. Step 2: Divide π by |7|. Step 3: The period is 1/7π.

Secant and Cosecant

Objective: Explain secant and cosecant, apply the rule, and justify relevant restrictions.

Secant is 1/cos x and cosecant is 1/sin x, wherever the denominators are nonzero. Their graphs have vertical asymptotes at the zeros of the corresponding denominator. For the untransformed functions, output lies in (−∞,−1]∪[1,∞). A reciprocal operation is different from an inverse function: sec x is not arccos x.

Formulas

$sec x=1/cos x; csc x=1/sin x$

Worked examples

Try itIf cos θ=3/10, find sec θ.Show answer

Step 1: Use sec θ=1/cos θ. Step 2: Invert 3/10; it is nonzero. Step 3: sec θ=10/3.

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