MIND ACTION SERIES
With the Educators, for the Educators
MATHEMATICS WORKSHOP 2022
TRIGONOMETRIC FUNCTIONS
GRADE 11
Presented by: Jurg Basson
Attending this Workshop = 5 SACE Points
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Trigonometric Functions
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In Grade 10 we learnt to sketch graphs to represent the values of trigonometric ratios for different
angles.
The three basic graphs are:
y = tan x
y = cos x
y = sin x
y
y
y
1
1
1
90°
180°
270°
90°
360°
180°
−1
−1
x
x
270° 360°
x
45°
90°
135° 180°
−1
We also learnt some definitions:
Period:
The length of the interval that contains exactly one cycle of the graph.
Amplitude:
1
[maximum y-value − minimum y-value]
2
The basic properties of y = sin x, y = cos x and y = tan x are summarised below:
y = sin x
y = cos x
y = tan x
Range
y ∈[−1;1]
OR
−1 ≤ y ≤ 1
y ∈[−1;1]
OR
−1 ≤ y ≤ 1
y ∈!
Amplitude
1
(1− (−1))
2
=1
1
(1− (−1))
2
=1
N.A.
tan doesn’t have a
maximum or minimum
y-value
Period
360°
360°
180°
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AMPLITUDE CHANGES AND VERTICAL SHIFTS
The role of a and q
In Grade 10 we studied some transformations of the trigonometric functions. Our functions were
defined as
y = asin x + q
y = a cos x + q
y = a tan x + q
• When the numerical value of a increases (ignoring the sign), the amplitude of the sin and cos
graphs increase. The numerical value of a determines the steepness of the tan graph.
• If a is negative, the graphs are all reflected across the x-axis.
• q shifts the graphs up (q > 0) or down (q < 0).
EXAMPLE 1
Consider the function f (x) = 2sin x − 1.
(a)
Sketch the graph of f for x ∈[0°; 360°].
(b)
Write down the period and amplitude of f .
(c)
Write down the range of f.
SKETCHING FOR DIFFERENT INTERVALS
Thus far, we have sketched trigonometric functions for the interval [0°; 360°]. We can sketch these
functions for different intervals:
EXAMPLE 2
Consider the function g(x) = − cos x + 1.
(a)
Sketch the graph of g for x ∈[−360°; 360°].
(b)
Write down the period and amplitude of g.
(c)
Write down the range of g.
EXAMPLE 3
Consider the function h(x) = −3tan x + 2.
(a)
Sketch the graph of h for x ∈[−180°;90°].
(c)
Write down the range of h.
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(b)
Write down the period of h.
HORIZONTAL SHIFTS
We know that adding a constant to x (or subtracting a constant from x) in a function, shifts the graph
horizontally (left or right).
For example, y = sin(x + 30°) is obtained by shifting y = sin x 30° to the left and y = cos(x − 20°)
is obtained by shifting y = cos x 20° to the right.
The value added to, or subtracted from, the x in a function is called the p-value.
The role of p
Our three trigonometric functions can now be written as
i y = asin(x + p) + q
i y = a cos(x + p) + q
i y = a tan(x + p) + q
p shifts the graph horizontally:
p>0:
p<0:
The graph is shifted to the left.
The graph is shifted to the right.
EXAMPLE 4
Sketch the graphs of the following trigonometric functions:
(a)
y = sin(x + 30°); x ∈[−390°; 330°]
(b)
y = tan(x − 15°); x ∈[−75°;195°]
EXAMPLE 5
Sketch the graphs of the following functions:
(a)
y = −2 cos(x − 20°); x ∈[−160°;200°]
(b)
y = tan(x + 60°) − 1; x ∈[−150°;180°]
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PERIOD CHANGES
Let us consider y = sin 2x.
Using a table:
x
0°
45°
y
0
1
90°
135°
180°
225°
270°
315°
360°
−1
0
1
0
−1
0
360°
x
0
y
1
45°
90°
135°
180° 225° 270°
315°
−1
From the sketch we see that the period of y = sin 2x is 180° (half of 360°).
The value multiplied by x (the coefficient of x) is called the k-value: y = sin kx.
The role of k
Our three trigonometric functions can now be written as:
i y = asin k(x + p) + q
i y = a cos k(x + p) + q
i y = a tan k(x + p) + q
k affects the period of the graph:
For the sin- and cos graphs, the period becomes
For the tan graph, the period becomes
360°
.
k
180°
.
k
EXAMPLE 6
Sketch the graphs of each of the following functions:
(a)
y = sin 3x; x ∈[0°; 360°]
(b)
1
y = cos x; x ∈[−360°; 360°]
2
(c)
y = tan 2x; x ∈[−180°;180°]
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EXAMPLE 7
Sketch the graphs of the following functions:
(a)
y = 2sin 2x − 1; x ∈[0°; 360°]
(b)
y = cos 2(x − 15°); x ∈[−120°;150°]
(c)
y = tan(3x + 30°); x ∈[−25°;50°]
FINDING THE EQUATION OF A TRIGONOMETRIC FUNCTION
EXAMPLE 8
Determine the equations of the following trigonometric functions:
(a)
y
2
180° x
90°
−2
(b)
y
−150°
−60°
30°
120°
210°
x
−1
−2
(c)
y
1
−35°
10° 32,5°
55°
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x
GRAPH INTERPRETATION
EXAMPLE 9
3
The functions f (x) = −2sin(x + p); − 30° ≤ x ≤ 180° and g(x) = cos x; − 30° ≤ x ≤ 180° are
2
sketched below. A is a turning point of f and B is a turning point of g and a point of intersection
between f and g.
y
f
−30°
C
180°
g
x
B
A
(a)
Write down the value of p.
(b)
What is the minimum value of f (x)?
(c)
What is the period of g?
(d)
What is the amplitude of f ?
(e)
Write down the coordinates of A and B.
(f)
Determine the range of h if h(x) =
(g)
1
f (x) − 1
2
Write down the equation of m if m is obtained by shifting g 15° to the right and 2 units up.
(h)
For which value(s) of x is
(i)
(i)
f (x) = g(x)?
(ii)
f (x) > g(x)?
(iii)
f (x)⋅ g(x) ≤ 0?
For which values of k will g(x) = k have one solution?
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