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Trigonometric Functions: Changes in Period
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This document deals with the changes in the period of trigonometric functions.
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Grade 11 Trigonometric FunctionsDocument 3: Period changes (the role of 𝑘)
Remember that the period of a graph is the number of degrees needed to complete one cycle of a
pattern (one complete wave).
For the basic graphs: sin 𝑥 has a period of 360°
cos 𝑥 has a period of 360°
tan 𝑥 has a period of 180°
First, let us consider 𝑦 = sin 2𝑥
Using a calculator and table:
𝑥 0° 45° 90° 135° 180° 225° 270° 315° 360°
𝑦 0 1 0 -1 0 1 0 -1 0
𝑔(𝑥) = sin 2𝑥 𝑓(𝑥) = sin 𝑥
From the sketch we can see that the period of 𝑦 = sin 2𝑥 is 180° (half of the basic sin graph’s period of
360°).
The value multiplied by 𝑥 (the coefficient of 𝑥) is commonly indicated using the letter k. (but it doesn't
have to be 𝑘 )
𝑦 = sin 𝑘𝑥
Multiplying the 𝑥 in the equation by a constant other than 1 changes the original period of the graph:
360°
e.g. In 𝑦 = sin 2𝑥 the 𝑥 is multiplied by 2, so the period becomes = 180°
2
1 1 360°
e.g. In 𝑦 = cos 3 𝑥 the 𝑥 is multiplied by 3 , so the period becomes 1 𝑜𝑟 360° × 3 = 1080°
3
360°
For the sin and cos graphs, the period becomes 𝑘
180°
For the tan graph, the period becomes , since the period of a tan graph is 180°.\
𝑘
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