6.03 Periodicity

It is important to know the sign (positive or negative) that each function has in the four different quadrants. We can use this information to quickly find the right trigonometric expression using our reference angle.

Remember that \sin\theta is equal to the y-coordinate (height), \cos\theta is equal to the x-coordinate (length), and \tan\theta is the gradient of the line from the origin to the point. Using this information we can deduce the sign for each function in each quadrant:

Second quadrant:

• y is positive: \sin \theta positive

• x is negative: \cos \theta negative

• Gradient is negative: \tan \theta negative

Third quadrant:

• y is negative: \sin \theta negative

• x is negative: \cos \theta negative

• Gradient is positive: \tan \theta positive

Fourth quadrant:

• y is negative: \sin \theta negative

• x is positive, \cos \theta positive

• Gradient is negative: \tan \theta negative

We can see this pattern of signs reflected in the graphs for these functions:

Sine is positive in the first and second quadrants.

Tangent is positive in the first and third quadrants.

Cosine is positive in the first and fourth quadrants.

Here is the same information summarised on the unit circle.

You might be taught a mnemonic such as "All Stations To Central" to help you remember this sign information.

Now that we can determine the sign of a ratio based on its quadrant, we can use the symmetry of the circle to find equivalent angles - angles with the same trigonometric ratio, up to their sign.

The relative acute angle, or reference angle, is always between 0\degree and 90\degree. It is shown here in quadrant 1.

Let's start by finding them for angles in the second quadrant:

The height of the two points are equal, which means:\sin \theta =\sin \left(180\degree -\theta \right)

The horizontal lengths are equal, but opposite in sign, which means:-\cos \theta =\cos \left(180\degree -\theta \right)

The gradient of the two lines are equal in magnitude but opposite in sign, which means:-\tan \theta =\tan \left(180\degree -\theta \right)

By combining the relative acute angle with our sign information, we can continue like this for the other quadrants. This is summarised in the table below:

Examples

Example 1

Write \sin 147\degree using an acute angle.

Worked Solution

Create a strategy

Consider which quadrant the angle is in.

Apply the idea

147\degree is between 90\degree and 180\degree so it is in the second quadrant.

We know that sine is positive in the second quadrant because y is positive.

This diagram shows that an acute angle 180°−\theta is related to \theta in the second quadrant.

180\degree-147\degree=33\degree which is the reference angle.

\displaystyle \sin \theta	\displaystyle =	\displaystyle + \sin \left(33\degree \right)	Use the reference angle
	\displaystyle =	\displaystyle \sin 33\degree	Evaluate

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Example 2

For \sin 300\degree , find the equivalent trigonometric expression in the first quadrant.

Worked Solution

Create a strategy

Find the quadrant the angle lies in and determine the sign of the ratio.

Apply the idea

Starting at the positive x-axis and rotating 300\degree anticlockwise, we move into the fourth quadrant.

This diagram shows that an angle of 360\degree-\theta in the fourth quadrant is equivalent to an acute angle of \theta in the first quadrant for the trig ratios.

Sine is negative in the fourth quadrant.

\displaystyle \sin 300\degree	\displaystyle =	\displaystyle -\sin (360\degree -300\degree)	Find the acute angle and use the sign
	\displaystyle =	\displaystyle -\sin 60\degree	Evaluate the angle

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What can we say about angles larger than 360\degree, or angles with a negative value?

A rotation of more than 360\degree means we have completed more than a full lap around the unit circle. But when we're finished rotating we will land on a point that we had already passed over during the first rotation.

A negative rotation happens in the opposite direction, clockwise rather than anti-clockwise, stopping in a position that matches some equivalent positive rotation between 0\degree and 360\degree.

We can rotate as much as we like in either direction, with each ratio changing in a predictable and repeating way - we use the word periodic to capture this property - and land on some point. The smallest amount of positive rotation to reach that point will produce the same values for the trigonometric ratios, and we can then use everything we learned before to find what we're looking for.

In practice we add or subtract multiples of 360\degree to obtain an angle between 0\degree and 360\degree. After determining what quadrant the angle lies in, we proceed as before.

Examples