Everything Maths Grade 11

CHAPTER 17. TRIGONOMETRY 17.2

Functions of the Formy=sin(θ+p) EMBCW

In the equation, y = sin(θ + p), p is a constant and has different effects on the graph of the function.
The general shape of the graph of functions ofthis form is shown in Figure 17.4 for the function
f(θ) = sin(θ + 30◦).

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-210-240-270-300-330 -30-60-90-120-150-180 30 60 90 120 150 180 210 240 270 300 330

Figure 17.4: Graph of f(θ) = sin(θ + 30◦) (solid line) and the graph of g(θ) = sin(θ) (dotted line).

Exercise 17 - 4

On the same set of axes, plot the following graphs:

1. a(θ) = sin(θ− 90 ◦)


2. b(θ) = sin(θ− 60 ◦)


3. c(θ) = sin θ


4. d(θ) = sin(θ + 90◦)


5. e(θ) = sin(θ + 180◦)

Use your results to deduce the effect of p.

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You should have foundthat the value of p affects the position ofthe graph along the y-axis (i.e. the
y-intercept) and the position of the graph along the x-axis (i.e. the phase shift). The p value shifts the
graph horizontally. If p is positive, the graph shifts left and if p is negative the graph shifts right.

These different properties are summarised in Table 17.4.

Domain and Range

For f(θ) = sin(θ + p), the domain is{θ : θ∈R} because there is no value of θ∈R for which f(θ) is
undefined.

The range of f(θ) = sin(θ + p) is{f(θ) : f(θ)∈ [−1;1]}.