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7.03 Periodicity

Lesson

Introduction

Graphs of trigonometric functions extend along the x-axis in both directions. This has concrete meaning when considering angles of rotation around the unit circle - these rotations can be any amount, both positive (anti-clockwise rotations) and negative (clockwise rotations).

For function values of angles greater than 90\degree, we can always find equivalent trigonometric expressions using acute reference angles.

Sign in quadrants

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:

The unit circle with an angle theta in the second quadrant. Ask your teacher for more information.

Second quadrant:

  • y is positive: \sin \theta positive

  • x is negative: \cos \theta negative

  • Gradient is negative: \tan \theta negative

The unit circle with an angle theta in the thord quadrant. Ask your teacher for more information.

Third quadrant:

  • y is negative: \sin \theta negative

  • x is negative: \cos \theta negative

  • Gradient is positive: \tan \theta positive

The unit circle with an angle theta in the fourth quadrant. Ask your teacher for more information.

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:

A graph of the sine function with positive values highlighted. Ask your teacher for more information.
Sine is positive in the first and second quadrants.
A graph of the tangent function with positive values highlighted. Ask your teacher for more information.
Tangent is positive in the first and third quadrants.
A graph of the cosine function with positive values highlighted. Ask your teacher for more information.
Cosine is positive in the first and fourth quadrants.
The unit circle with the sines of the trigonometric ratios. Ask your teacher for more information.
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.

A symmetry of the circle with the same trigonometric ratio . Ask your teacher for more 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:

A diagram shows the relative acute angle. Ask your teacher for more information.

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:

First quadrantSecond quadrantThird quadrantFourth quadrant
\text{Angle }\theta \\ \sin \theta \text{ is positive} \\ \cos \theta \text{ is positive} \\ \tan \theta \text{ is positive}\text{Angle } 180 \degree - \theta \\ \sin (180\degree - \theta) = \sin \theta \\ \cos (180\degree - \theta) = -\cos \theta \\ \tan (180\degree - \theta) = -\tan \theta \text{Angle } 180 \degree + \theta \\ \sin (180\degree + \theta) = -\sin \theta \\ \cos (180\degree + \theta) = -\cos \theta \\ \tan (180\degree + \theta) = \tan \theta \text{Angle } 360 \degree - \theta \\ \sin (360\degree - \theta) = -\sin \theta \\ \cos (360\degree - \theta) = \cos \theta \\ \tan (360\degree - \theta) = -\tan \theta

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
This image shows the signs of the trigonometric ratios in each quadrant. Ask your teacher for more information.

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\degreeEvaluate

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
A diagram shows the acute angle in the fourth quadrant equals to the angle in first quadrant. Ask your teacher for more information.

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
Idea summary

The trigonometric ratio of any angle can be reduced to the same ratio applied to its acute reference angle (between 0\degree and 90\degree).

The sign of any trigonometric ratio can be determined from its quadrant, using the ASTC mnemonic.

Angles larger than 360\degree or less than 0\degree can be related to angles between 0\degree and 360\degree by adding or subtracting multiples of 360\degree.

Negative angles and angles larger than 360

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

Example 3

Write \cos 605\degree using an acute angle.

Worked Solution
Create a strategy

Subtract 360\degree from the angle.

Apply the idea
\displaystyle 605\degree -360\degree \displaystyle =\displaystyle 245\degreeSubtract 360\degree

The angle 245\degree is in the third quadrant, where cosine is negative. To find the equivalent acute angle we can subtract 180 \degree from 245\degree .

\displaystyle \cos 605\degree \displaystyle =\displaystyle -\cos (245 \degree - 180\degree )Find the acute angle and use the sign
\displaystyle =\displaystyle -\cos 65\degree Simplify
Idea summary

If we start with a negative angle or an angle larger than 360 \degree, 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.

Outcomes

VCMMG368 (10a)

Use the unit circle to define trigonometric functions, and graph them with and without the use of digital technologies.

VCMMG369 (10a)

Solve simple trigonometric equations.

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