Everything Maths Grade 11

17.2 CHAPTER 17. TRIGONOMETRY

2. b(θ) = tan1θ


3. c(θ) = tan1, 5 θ


4. d(θ) = tan2θ


5. e(θ) = tan2, 5 θ

Use your results to deduce the effect of k.

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You should have foundthat, once again, the value of k affects the periodicity (i.e. frequency) of the
graph. As k increases, the graph is more tightly packed. As k decreases, the graph ismore spread out.
The period of the tan graph is given by^180

◦

k.

These different properties are summarised in Table 17.3.

Table 17.3: Table summarising general shapes and positions of graphs of functions of theform
y = tan(kθ).

k > 0 k < 0

Domain and Range

For f(θ) = tan(kθ), the domain of one branch is{θ : θ∈ (−^90

◦

k;

90 ◦
k)} because the function is
undefined for θ =−^90
◦
k and θ =

90 ◦

k.

The range of f(θ) = tan(kθ) is{f(θ) : f(θ)∈ (−∞;∞)}.

Intercepts

For functions of the form, y = tan(kθ), the details of calculating the intercepts with the x and y axis
are given.

There are many x-intercepts; each one ishalfway between the asymptotes.

The y-intercept is calculated as follows:

y = tan(kθ)

yint = tan(0)

= 0

Asymptotes

The graph of tan kθ has asymptotes becauseas kθ approaches 90 ◦, tan kθ approaches infinity. In other
words, there is no defined value of the functionat the asymptote values.