Evaluate Expressions with Identities Revision Notes for HSC SSCE Mathematics Advanced

Evaluate Expressions with Identities

Introduction

In this topic, you will learn how to evaluate trigonometric expressions using complementary angle identities and quadrant rules. This skill allows you to find exact values for trigonometric functions of any angle by breaking down complex expressions into simpler, manageable parts.

After studying this note, you will be able to:

Evaluate trigonometric expressions involving acute angles using complementary angle identities
Find the exact value of trigonometric functions for angles of any magnitude using reference angles and quadrant signs
Combine complementary angle identities and quadrant rules to evaluate complex trigonometric expressions

Expressions with acute angles

Acute angles are angles between $0°$ and $90°$ . When working with trigonometric expressions involving acute angles, you can simplify them using complementary angle identities and the exact values of standard angles like $30°$ , $45°$ , and $60°$ .

A key technique is to rewrite angles as the difference from $90°$ . For example, the angle $60°$ can be expressed as $90° - 30°$ , which allows you to apply complementary angle identities.

Complementary angle identities

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These identities relate trigonometric functions of complementary angles (angles that add to $90°$ ):

$\tan(90° - \theta) = \cot \theta$
$\cos(90° - \theta) = \sin \theta$
$\sin(90° - \theta) = \cos \theta$

Additionally, remember that the cotangent function can be expressed as:

$\cot \theta = \frac{1}{\tan \theta}$

Worked example: evaluating acute angle expressions

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Worked Example: Evaluating Acute Angle Expressions

Problem: Without simplifying inside the brackets, evaluate $\tan(90° - 45°) + \cos(90° - 60°)$ in exact form.

Strategy: Apply complementary angle identities to rewrite each term, then use exact values for $45°$ and $60°$ .

Solution:

For $\tan(90° - 45°)$ :

\tan(90° - 45°) = \cot 45°

= \frac{1}{\tan 45°}

= \frac{1}{1}

= 1

For $\cos(90° - 60°)$ :

\cos(90° - 60°) = \sin 60°

= \frac{\sqrt{3}}{2}

Combining the results:

$\tan(90° - 45°) + \cos(90° - 60°) = 1 + \frac{\sqrt{3}}{2}$

chatImportant

Key Point: Complementary angle identities simplify expressions with acute angles by converting between trigonometric functions. Use exact values for standard angles to evaluate the resulting expressions.

Angles of any magnitude

To evaluate trigonometric expressions for angles of any magnitude (not just acute angles), you need to use the reference angle and the quadrant to determine the sign of the function.

The reference angle is the acute angle formed with the $x$ -axis. It tells you which standard angle to use for evaluation.

Quadrant sign rules

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For an angle $\theta$ in different quadrants:

Quadrant 1 ( $0° < \theta < 90°$ ): All functions are positive
Quadrant 2 ( $90° < \theta < 180°$ ): Only $\sin \theta$ and $\text{cosec } \theta$ are positive
Quadrant 3 ( $180° < \theta < 270°$ ): Only $\tan \theta$ and $\cot \theta$ are positive
Quadrant 4 ( $270° < \theta < 360°$ ): Only $\cos \theta$ and $\sec \theta$ are positive

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Memory aid: Remember "All Sin Tan Cos" going anticlockwise from Quadrant 1.

Example: $\sin 150° = \sin(180° - 30°) = \sin 30°$ (positive in Quadrant 2)

Exploration questions

To deepen your understanding, consider:

How does the quadrant affect the sign of $\cos 210°$ ?
Find the reference angle and determine if the result is positive or negative.

Worked example: angles in any quadrant

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Worked Example: Angles in Any Quadrant

Problem: Evaluate $\cos 135°$ .

Strategy: Find the reference angle of $135°$ and apply the appropriate sign based on the quadrant where $135°$ is located.

Solution:

The angle $135°$ is in Quadrant 2, so:

\cos 135° = \cos(180° - 135°) \quad \text{Subtract } \theta \text{ from } 180° \text{ for the reference angle}

= \cos 45° \quad \text{Evaluate the subtraction}

= -\cos 45° \quad \text{Use the fact that } \cos \theta \text{ is negative in Quadrant 2}

= -\frac{\sqrt{2}}{2} \quad \text{Evaluate the exact value of } \cos 45°

chatImportant

Key Point: For angles of any magnitude, find the reference angle and use the quadrant to determine the sign of the trigonometric function. Express angles in terms of standard angles for evaluation.

Combining identities and quadrants

Complex trigonometric expressions may require you to combine complementary angle identities with quadrant adjustments for angles of any magnitude.

Steps to evaluate complex expressions

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Follow these steps systematically:

Apply complementary angle identities to simplify the expression
Determine the quadrant of the resulting angle
Use the reference angle and quadrant sign to evaluate

Worked example: combining techniques

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Worked Example: Combining Techniques

Problem: Without simplifying inside the brackets, evaluate $\sin(90° - 150°) + \cos 210°$ .

Strategy: Use complementary angle identities for the first term, find the reference angle and quadrant for both terms, and evaluate using exact values.

Solution:

For $\sin(90° - 150°)$ :

\sin(90° - 150°) = \cos 150°

Apply the identity $\sin(90° - \theta) = \cos \theta$ with $\theta = 150°$ .

= \cos(180° - 150°)

Subtract $\theta$ from $180°$ to find the reference angle.

= \cos 30°

Evaluate the subtraction.

= -\cos 30°

$\cos \theta$ is negative in Quadrant 2.

= -\frac{\sqrt{3}}{2}

Evaluate the exact value of $\cos 30°$ .

For $\cos 210°$ :

\cos 210° = \cos(210° - 180°)

Subtract $180°$ from $\theta$ to find the reference angle.

= \cos 30°

Evaluate the subtraction.

= -\cos 30°

$\cos \theta$ is negative in Quadrant 3.

= -\frac{\sqrt{3}}{2}

Evaluate the exact value of $\cos 30°$ .

Combining the results:

\sin(90° - 150°) + \cos 210° = -\frac{\sqrt{3}}{2} + \left(-\frac{\sqrt{3}}{2}\right)

Combine the results.

= -\frac{2\sqrt{3}}{2}

Evaluate the addition.

= -\sqrt{3}

Simplify.

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Key Point: Combine complementary angle identities with quadrant adjustments to evaluate complex trigonometric expressions. Simplify using identities, find reference angles, and apply the correct sign based on the quadrant.

Remember!

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Key Points to Remember:

Complementary angle identities allow you to convert between trigonometric functions for angles that add to $90°$ . The most common are $\sin(90° - \theta) = \cos \theta$ , $\cos(90° - \theta) = \sin \theta$ , and $\tan(90° - \theta) = \cot \theta$ .
Reference angles are acute angles formed with the $x$ -axis. Use them to evaluate trigonometric functions for angles of any magnitude by relating them to standard angles ( $30°$ , $45°$ , $60°$ ).
Quadrant rules determine the sign of trigonometric functions: All positive in Q1, only sin positive in Q2, only tan positive in Q3, only cos positive in Q4.
Exact values for standard angles must be memorised: $\sin 30° = \frac{1}{2}$ , $\cos 30° = \frac{\sqrt{3}}{2}$ , $\tan 45° = 1$ , $\sin 60° = \frac{\sqrt{3}}{2}$ , $\cos 45° = \frac{\sqrt{2}}{2}$ .
When evaluating complex expressions, work systematically: apply identities first, determine quadrants, find reference angles, then evaluate with correct signs.

Evaluate Expressions with Identities (HSC SSCE Mathematics Advanced): Revision Notes