Trigonometric Functions: Changes in Period

Grade 11 Trigonometric Functions

Document 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