Related Angles and Identities Revision Notes for HSC SSCE Mathematics Advanced

Related Angles and Identities

Introduction to related angles

When working with trigonometric ratios, we often need to find values for angles of any size. A related angle (also called a reference angle) helps us do this. The related angle for any angle $\theta$ is the acute angle formed between the terminal ray and the $x$ -axis on the unit circle.

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This concept is incredibly useful because it allows us to express trigonometric ratios for any angle using an acute angle, which we can more easily calculate or look up in tables. Instead of memorizing values for hundreds of angles, we only need to know the ratios for acute angles and apply the appropriate rules!

Understanding quadrants

The coordinate plane is divided into four quadrants by the $x$ -axis and $y$ -axis. Each quadrant contains angles within a specific range.

The four quadrants are classified as:

Quadrant 1: $0° < \theta < 90°$
Quadrant 2: $90° < \theta < 180°$
Quadrant 3: $180° < \theta < 270°$
Quadrant 4: $270° < \theta < 360°$

Understanding which quadrant an angle lies in is essential for determining the signs of its trigonometric ratios.

Finding related angles

The method for finding a related angle depends on which quadrant the angle is in. Here are the formulas for each quadrant:

Quadrant 1: Related angle $= \theta$
Quadrant 2: Related angle $= 180° - \theta$
Quadrant 3: Related angle $= \theta - 180°$
Quadrant 4: Related angle $= 360° - \theta$

chatImportant

If your angle is outside the range $0°$ to $360°$ , you first need to bring it into this range by adding or subtracting multiples of $360°$ . This is a crucial first step that students often forget!

The ASTC rule

The ASTC rule (sometimes called the CAST rule) is a memory aid that tells us which trigonometric functions are positive in each quadrant. The letters stand for the positive functions in each quadrant, starting from Quadrant 1 and moving anticlockwise.

Here's what ASTC means:

Quadrant 1 (A): All trigonometric ratios are positive ( $\sin \theta$ , $\cos \theta$ , and $\tan \theta$ are all positive)
Quadrant 2 (S): Only sine is positive ( $\sin \theta > 0$ , while $\cos \theta < 0$ and $\tan \theta < 0$ )
Quadrant 3 (T): Only tangent is positive ( $\tan \theta > 0$ , while $\sin \theta < 0$ and $\cos \theta < 0$ )
Quadrant 4 (C): Only cosine is positive ( $\cos \theta > 0$ , while $\sin \theta < 0$ and $\tan \theta < 0$ )

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Memory tip: "All Stations To Central" or "All Students Take Calculus" can help you remember the order. Choose whichever phrase resonates with you and practice it until it becomes automatic!

Related angle identities

When angles are expressed in the forms $180° \pm A$ or $360° - A$ , where $A$ is an acute angle, we can use specific identities to find their trigonometric ratios. These identities are summarised in the table below:

Angle form	$\sin \theta$	$\cos \theta$	$\tan \theta$
$180° - A$	$\sin A$	$-\cos A$	$-\tan A$
$180° + A$	$-\sin A$	$-\cos A$	$\tan A$
$360° - A$	$-\sin A$	$\cos A$	$-\tan A$

These identities allow us to express the trigonometric ratios of any angle in terms of an acute angle $A$ , making calculations much simpler. Notice the patterns in the signs - they follow directly from the ASTC rule for each quadrant.

Negative angle identities

Negative angles are formed by rotating clockwise from the positive $x$ -axis, rather than anticlockwise. When we have a negative angle $-\theta$ , we can find its trigonometric ratios using these important identities.

The diagram above shows that for an angle $\theta = 60°$ , the point on the unit circle is $(\cos 60°, \sin 60°)$ . For the negative angle $-\theta = -60°$ , the point is $(\cos 60°, -\sin 60°)$ . Notice that the x-coordinate stays the same, but the y-coordinate becomes negative.

This observation leads to the following fundamental identities:

$\cos(-\theta) = \cos \theta$

$\sin(-\theta) = -\sin \theta$

$\tan(-\theta) = \frac{\sin(-\theta)}{\cos(-\theta)} = \frac{-\sin \theta}{\cos \theta} = -\tan \theta$

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These identities show that:

Cosine is an even function (unchanged for negative angles)
Sine and tangent are odd functions (they change sign for negative angles)

Understanding these properties helps you work with negative angles without confusion!

Worked examples

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Worked Example 1: Finding a related angle and using it to evaluate cosine

Question: Find the related angle for $750°$ and express $\cos 750°$ using an acute angle.

Solution:

First, we need to reduce $750°$ to an angle between $0°$ and $360°$ :

$750° = 750° - 360° - 360°$

$750° = 30°$

The angle $30°$ is in Quadrant 1, where $\cos \theta = \cos A$ and the related angle is $30°$ .

Therefore:

$\cos 750° = \cos 30°$

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Worked Example 2: Evaluating sine without technology

Question: Evaluate $\sin 300°$ without using a calculator.

Solution:

The angle $300°$ is in Quadrant 4 (since $270° < 300° < 360°$ ).

To find the related angle, we apply the formula $\theta = 360° - A$ :

$\theta = 360° - A$

$300° = 360° - A$

$A = 60°$

In Quadrant 4, sine is negative (using the ASTC rule). We know that $\sin 60° = \frac{\sqrt{3}}{2}$ .

Therefore:

$\sin 300° = -\sin 60°$

$\sin 300° = -\frac{\sqrt{3}}{2}$

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Worked Example 3: Using a related angle to find tangent

Question: Express $\tan 195°$ using an acute angle and determine its value, rounded to two decimal places.

Solution:

The angle $195°$ is in Quadrant 3 (since $180° < 195° < 270°$ ).

To find the related angle, we use the formula $\theta = 180° + A$ :

$\theta = 180° + A$

$195° = 180° + A$

$A = 15°$

In Quadrant 3, tangent is positive (using the ASTC rule).

Therefore:

$\tan 195° = \tan 15°$

$\tan 195° = 0.27 \text{ (rounded to two decimal places)}$

Note: The exact value of $\tan 15°$ can be found using compound angle formulas, but for this example, we use the identity to show that $\tan 195°$ has the same value as $\tan 15°$ and is positive.

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Worked Example 4: Evaluating negative angles using identities

Question: Find $\sin(-60°)$ and $\cos(-60°)$ using identities.

Solution:

We apply the negative angle identities $\sin(-\theta) = -\sin \theta$ and $\cos(-\theta) = \cos \theta$ .

We know that $\sin 60° = \frac{\sqrt{3}}{2}$ and $\cos 60° = \frac{1}{2}$ .

For sine:

$\sin(-60°) = -\sin 60°$

$\sin(-60°) = -\frac{\sqrt{3}}{2}$

For cosine:

$\cos(-60°) = \cos 60°$

$\cos(-60°) = \frac{1}{2}$

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Worked Example 5: Finding tangent of a negative angle

Question: Express $\tan(-150°)$ using a positive acute angle and determine its sign.

Solution:

First, we convert $-150°$ to a positive angle:

$-150° + 360° = 210°$

The angle $210°$ is in Quadrant 3 (since $180° < 210° < 270°$ ).

To find the related angle, we use the formula $\theta = 180° + A$ :

$\theta = 180° + A$

$210° = 180° + A$

$A = 30°$

In Quadrant 3, tangent is positive (using the ASTC rule).

Therefore:

$\tan(-150°) = \tan 30°$

So $\tan(-150°) = \tan 30°$ is positive.

Alternatively, we could use the negative angle identity directly:

$\tan(-150°) = -\tan 150°$

Since $150°$ is in Quadrant 2 where tangent is negative, we have $\tan 150° = -\tan 30°$ .

Therefore: $\tan(-150°) = -(-\tan 30°) = \tan 30°$ , which is positive.

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Key Points to Remember:

Related angles are acute angles between the terminal ray and the $x$ -axis, used to find trigonometric ratios for any angle.
Use the ASTC rule to determine which trigonometric functions are positive in each quadrant: All (Q1), Sine (Q2), Tangent (Q3), Cosine (Q4).
Related angle formulas vary by quadrant: $\theta$ (Q1), $180° - \theta$ (Q2), $\theta - 180°$ (Q3), $360° - \theta$ (Q4).
Identity table for $180° \pm A$ and $360° - A$ helps express any angle's trigonometric ratios using acute angles.
Negative angle identities: $\cos(-\theta) = \cos \theta$ , $\sin(-\theta) = -\sin \theta$ , $\tan(-\theta) = -\tan \theta$ .

Related Angles and Identities (HSC SSCE Mathematics Advanced): Revision Notes