Worksheet 4.8 Properties of Trigonometric Functions

1○ These trig functions are periodic- they repeat themselves after a certain period.
• sin x and cos x have period 2π:
• tan(x) has period π:
i.e. sin x = sin(x + 2π) ∀x ∈ R
cos x = cos(x + 2π) ∀x ∈ R
i.e. tan(x) = tan(x + π) ∀x ∈ R
2○ Note that sin x and cos x both lie between −1 and 1.
3○ Note that tan x is undefined for x = (2k−1)π
2 , k ∈ Z.
4○ From the graphs we can see that
sin x + π
2
π
cos − x
2
5○ Since sin x and tan x are odd functions we have
Since cos x is an even function we have
= cos x
= sin x
sin(−x) = − sin x ∀x ∈ R
tan(−x) = − tan x ∀x ∈ R
cos(−x) = cos x ∀x ∈ R
These graphs alter if we change the the period or amplitude, or if there is a phase shift.
Consider the graph of y = sin x. In general we can think of this as y = A sin n(x − a), where
• A is the amplitude
• n alters the period (period = 2π
n )
• by subtracting a from x, the graph shifts to the right by a.
So y = sin x has amplitude 1 and period 2π.
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