Victorian Curriculum Year 10A - 2020 Edition
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7.03 Periodicity
Lesson

Graphs of trigonometric functions extend along the $x$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^\circ$90°, 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 $\sin\theta$sinθ is equal to the $y$y-coordinate (height), $\cos\theta$cosθ is equal to the $x$x-coordinate (length), and $\tan\theta$tanθ 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:

  • Height positive, $\sin\theta$sinθ positive
  • Length negative, $\cos\theta$cosθ negative
  • Gradient negative, $\tan\theta$tanθ negative

Third quadrant:

  • Height negative, $\sin\theta$sinθ negative
  • Length negative, $\cos\theta$cosθ negative
  • Gradient positive, $\tan\theta$tanθ positive

Fourth quadrant:

  • Height negative, $\sin\theta$sinθ negative
  • Length positive, $\cos\theta$cosθ positive
  • Gradient negative, $\tan\theta$tanθ negative
Unit circle showing $\sin\theta$sinθ, $\cos\theta$cosθ and $\tan\theta$tanθ in the three other quadrants.

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

Sine is positive in the first and second quadrants
Tan 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 itself:

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

 

Equivalent Angles

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^\circ$0° and $90^\circ$90°. Let's start by finding them for angles in the second quadrant:

The height of the two points are equal, which means:

$\sin\theta=\sin\left(180^\circ-\theta\right)$sinθ=sin(180°θ)

The horizontal lengths are equal, but opposite in sign, which means:

$-\cos\theta=\cos\left(180^\circ-\theta\right)$cosθ=cos(180°θ)

The gradient of the two lines are equal in magnitude but opposite in sign, which means:

$-\tan\theta=\tan\left(180^\circ-\theta\right)$tanθ=tan(180°θ)

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 quadrant

Angle $\theta$θ

$\sin\theta$sinθ is positive

$\cos\theta$cosθ is positive

$\tan\theta$tanθ is positive

Second quadrant

Angle $180^\circ-\theta$180°θ

$\sin(180^\circ-\theta)=\sin\theta$sin(180°θ)=sinθ

$\cos(180^\circ-\theta)=-\cos\theta$cos(180°θ)=cosθ

$\tan(180^\circ-\theta)=-\tan\theta$tan(180°θ)=tanθ

Third quadrant

Angle $180^\circ+\theta$180°+θ

$\sin(180^\circ+\theta)=-\sin\theta$sin(180°+θ)=sinθ

$\cos(180^\circ+\theta)=-\cos\theta$cos(180°+θ)=cosθ

$\tan(180^\circ+\theta)=\tan\theta$tan(180°+θ)=tanθ

Fourth quadrant

Angle $360^\circ-\theta$360°θ

$\sin(360^\circ-\theta)=-\sin\theta$sin(360°θ)=sinθ

$\cos(360^\circ-\theta)=\cos\theta$cos(360°θ)=cosθ

$\tan(360^\circ-\theta)=-\tan\theta$tan(360°θ)=tanθ

 

Angles outside of the domain $0^{\circ}\le\theta\le360^{\circ}$0θ360

What can we say about angles larger than $360^\circ$360°, or angles with a negative value?

A rotation of more than $360^\circ$360° 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^\circ$0° and $360^\circ$360°.

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^\circ$360° to obtain an angle between $0^\circ$0° and $360^\circ$360°. After determining what quadrant the angle lies in, we proceed as before.

 

Worked examples

Example 1

Express $\cos117^\circ$cos117° in terms of a first quadrant angle.

Think: The angle $117^\circ$117° is between $90^\circ$90° and $180^\circ$180°, so it is in the second quadrant. The point representing $117^\circ$117° on the unit circle diagram, where the radius cuts the circle, must have a negative horizontal coordinate. We should subtract $117^\circ$117° from $180^\circ$180° to find the relative acute angle.

Do: $\cos117^\circ=-\cos\left(180^\circ-117^\circ\right)=-\cos63^\circ$cos117°=cos(180°117°)=cos63°.

Example 2

Evaluate $\cos\left(240^\circ\right)$cos(240°) and find five other angles with the same cosine value.

Think: The angle $240^\circ$240° is between $180^\circ$180° and $270^\circ$270°, so it is in the third quadrant. The point representing $240^\circ$240° on the unit circle diagram, where the radius cuts the circle, must have a negative horizontal coordinate. We should subtract $180^\circ$180° from $240^\circ$240° to find the relative acute angle.

Do: $\cos\left(240^\circ\right)=-\cos(240^\circ-180^\circ)=-\cos(60^\circ)=-\frac{1}{2}$cos(240°)=cos(240°180°)=cos(60°)=12.

We can find an angle with the same cosine value by selecting the corresponding angle in the second quadrant, because its point on the unit circle will have the same (negative) length away from the origin. To find this angle we can subtract the relative acute angle $60^\circ$60° from $180^\circ$180° to find: $\cos\left(240^\circ\right)=\cos\left(180^\circ-60^\circ\right)=\cos\left(120^\circ\right)$cos(240°)=cos(180°60°)=cos(120°).

We can now find many more angles by adding or subtracting multiples of a full rotation ($360^\circ$360°) from either $240^\circ$240° or $120^\circ$120° - in fact, we can find infinitely many:

$\ldots=\cos\left(-240^\circ\right)=\cos\left(-120^\circ\right)=\cos\left(120^\circ\right)=\cos\left(240^\circ\right)=\cos\left(480^\circ\right)=\cos\left(600^\circ\right)=\ldots$=cos(240°)=cos(120°)=cos(120°)=cos(240°)=cos(480°)=cos(600°)=

Example 3

Express the sine, cosine and tangent functions of the angle $512^\circ$512° in terms of an angle in the first quadrant.

The angle $512^\circ$512° is more than once around the full circle. So, it is equivalent to $512^\circ-360^\circ=152^\circ$512°360°=152°, which is in the second quadrant. We subtract the angle from $180^\circ$180° to find:

$\sin512^\circ=\sin28^\circ$sin512°=sin28°

$\cos512^\circ=-\cos28^\circ$cos512°=cos28°

$\tan512^\circ=-\tan28^\circ$tan512°=tan28°

Remember!

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

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

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

 

Practice questions

Question 1

Consider the expression $\tan120^\circ$tan120°.

  1. In which quadrant is $120^\circ$120°?

    First quadrant

    A

    Second quadrant

    B

    Third quadrant

    C

    Fourth quadrant

    D

    First quadrant

    A

    Second quadrant

    B

    Third quadrant

    C

    Fourth quadrant

    D
  2. What positive acute angle is $120^\circ$120° related to?

  3. Is $\tan120^\circ$tan120° positive or negative?

    Positive

    A

    Negative

    B

    Positive

    A

    Negative

    B
  4. Rewrite $\tan120^\circ$tan120° in terms of its relative acute angle. You do not need to evaluate $\tan120^\circ$tan120°.

Question 2

Write the following trigonometric expression using an acute angle:

$\sin106^\circ$sin106°

Question 3

For each of the following, find the equivalent trigonometric expression in the first quadrant.

You do not need to evaluate the trigonometric expression.

  1. $\sin300^\circ$sin300°
  2. $\cos165^\circ$cos165°
  3. $\tan220^\circ$tan220°

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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