Chapter 04: Trigonometry – Part 1: Trigonometric Functions, Angles and Graphs

Mathematics Hip-Hop Academy | A-Level Pure Mathematics

Chapter 04: Trigonometry

Part 1: Trigonometric Functions, Angles and Graphs

Build a strong foundation in trigonometric ratios, angular measurement, the unit circle, special angles, quadrants and trigonometric graphs.

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1. Learning Objectives

By the end of this lesson, you should be able to:

• Define and use the six trigonometric ratios.
• Understand positive and negative angles.
• Identify the four quadrants and the signs of trigonometric ratios.
• Calculate exact values of standard angles.
• Determine reference angles.
• Sketch and interpret sine, cosine and tangent graphs.

2. Introduction to Trigonometry

Trigonometry is the branch of mathematics concerned with relationships between angles and sides of triangles. It also studies circular functions and their graphs.

It is used in geometry, surveying, navigation, engineering, physics and many other fields.

3. The Six Trigonometric Ratios

Consider a right-angled triangle containing an acute angle θ. Relative to θ, its sides are called the opposite, adjacent and hypotenuse.

Important: The hypotenuse is always opposite the right angle. The opposite and adjacent sides are identified relative to the angle being considered.

sin θ = Opposite / Hypotenuse
cos θ = Adjacent / Hypotenuse
tan θ = Opposite / Adjacent

The remaining three ratios are reciprocals:

cosec θ = 1 / sin θ = Hypotenuse / Opposite
sec θ = 1 / cos θ = Hypotenuse / Adjacent
cot θ = 1 / tan θ = Adjacent / Opposite

Worked Example 1

A right-angled triangle has sides 3 cm, 4 cm and 5 cm. Let θ be the angle opposite the side of length 3 cm. Find all six trigonometric ratios.

Solution:

Opposite = 3, adjacent = 4, hypotenuse = 5.

sin θ = 3/5; cos θ = 4/5; tan θ = 3/4

cosec θ = 5/3; sec θ = 5/4; cot θ = 4/3

4. Angles: Degrees and Direction

An angle measures the amount of rotation between two directions.

• Anticlockwise rotation is positive.
• Clockwise rotation is negative.
• One complete revolution is 360°.
• A half revolution is 180°.
• A quarter revolution is 90°.

For example, +60° represents an anticlockwise rotation of 60°, while −60° represents a clockwise rotation of 60°.

Angles differing by 360° have the same terminal arm. For example, 30°, 390° and −330° are coterminal angles.

5. The Unit Circle

The unit circle is a circle of radius 1 centred at the origin of a Cartesian coordinate system.

If a point P on the unit circle corresponds to an angle θ measured from the positive x-axis, its coordinates are:

(x, y) = (cos θ, sin θ)

Therefore, the horizontal coordinate gives the cosine and the vertical coordinate gives the sine.

This relationship allows trigonometric functions to be defined for angles of any size, including negative angles and angles greater than 360°.

6. The Four Quadrants and Signs

Quadrant	Angle range	Positive ratios
I	0° to 90°	All six
II	90° to 180°	Sine and cosecant
III	180° to 270°	Tangent and cotangent
IV	270° to 360°	Cosine and secant

A useful memory aid is ASTC: All, Sine, Tangent, Cosine. Starting in quadrant I and moving anticlockwise, this indicates which ratios are positive.

Worked Example 2

Evaluate sin 150°.

Solution: 150° is in quadrant II, where sine is positive.

Reference angle = 180° − 150° = 30°.

sin 150° = sin 30° = 1/2

Worked Example 3

Evaluate cos 225°.

Solution: 225° is in quadrant III, where cosine is negative.

Reference angle = 225° − 180° = 45°.

cos 225° = −cos 45° = −√2/2

7. Special Angles and Exact Values

The standard angles 0°, 30°, 45°, 60° and 90° occur frequently in examination questions. Their exact values should be memorised.

θ	0°	30°	45°	60°	90°
sin θ	0	1/2	√2/2	√3/2	1
cos θ	1	√3/2	√2/2	1/2	0
tan θ	0	1/√3	1	√3	Undefined

Warning: tan 90° is undefined because tan θ = sin θ / cos θ and cos 90° = 0. Division by zero is not defined.

8. Reference Angles

A reference angle is the acute positive angle formed between the terminal arm of an angle and the x-axis.

• Quadrant I: reference angle = θ.
• Quadrant II: reference angle = 180° − θ.
• Quadrant III: reference angle = θ − 180°.
• Quadrant IV: reference angle = 360° − θ.

Worked Example 4

Find the reference angle of 330° and evaluate tan 330°.

Reference angle = 360° − 330° = 30°.

Tangent is negative in quadrant IV.

tan 330° = −tan 30° = −1/√3

9. Graphs of Trigonometric Functions

9.1 The Sine Graph

The basic sine function is y = sin θ.

• Domain: all real values of θ.
• Range: −1 ≤ y ≤ 1.
• Period: 360°.
• Maximum value: 1.
• Minimum value: −1.

θ	0°	90°	180°	270°	360°
sin θ	0	1	0	−1	0

The sine graph begins at zero, rises to 1, returns to zero, falls to −1 and returns to zero after one complete cycle.

9.2 The Cosine Graph

The basic cosine function is y = cos θ.

• Domain: all real values of θ.
• Range: −1 ≤ y ≤ 1.
• Period: 360°.
• Maximum value: 1.
• Minimum value: −1.

θ	0°	90°	180°	270°	360°
cos θ	1	0	−1	0	1

The cosine graph begins at 1, falls to zero, reaches −1, rises to zero and returns to 1.

9.3 The Tangent Graph

The basic tangent function is y = tan θ.

• Domain: all real θ except θ = 90° + 180°k, where k is an integer.
• Range: all real numbers.
• Period: 180°.
• Vertical asymptotes: θ = 90° + 180°k.

θ	0°	45°	90°	135°	180°
tan θ	0	1	Undefined	−1	0

The tangent graph is not defined at 90° and 270° within one complete revolution. These positions are vertical asymptotes.

10. Examination Practice

Attempt all the questions before opening the answers.

1. Define the six trigonometric ratios using the sides of a right-angled triangle.
2. Find the six trigonometric ratios of an angle in a triangle whose opposite, adjacent and hypotenuse sides are 5, 12 and 13 respectively.
3. State the quadrant in which each angle lies: 120°, 210° and 315°.
4. Evaluate exactly: (a) sin 120°; (b) cos 240°; (c) tan 315°.
5. Find the reference angle of 225°.
6. State the period and range of y = sin θ.
7. State the period of y = tan θ and give its vertical asymptotes.
8. Explain why tan 90° is undefined.

11. Answers and Explanations

Open Answers 1–4

1. Sine = opposite/hypotenuse; cosine = adjacent/hypotenuse; tangent = opposite/adjacent. Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent respectively.

2.

sin θ = 5/13; cos θ = 12/13; tan θ = 5/12

cosec θ = 13/5; sec θ = 13/12; cot θ = 12/5

3. 120° is in quadrant II; 210° is in quadrant III; 315° is in quadrant IV.

4. (a) √3/2; (b) −1/2; (c) −1.

Open Answers 5–8

5. 225° − 180° = 45°.

6. Period = 360°; range = −1 ≤ y ≤ 1.

7. Period = 180°. Vertical asymptotes occur at θ = 90° + 180°k, where k is an integer.

8. tan θ = sin θ / cos θ. Since cos 90° = 0, tan 90° would require division by zero, which is undefined.

12. Part 1 Summary

• The six trigonometric ratios describe relationships between sides of a right-angled triangle.
• Anticlockwise angles are positive and clockwise angles are negative.
• The unit circle extends trigonometric definitions to angles of any magnitude.
• The quadrant determines the sign of each trigonometric ratio.
• Exact values of standard angles must be known.
• Sine and cosine have period 360°, while tangent has period 180°.

Preparation for Part 2: In the next lesson, we shall study trigonometric identities, including the Pythagorean identities, reciprocal identities and methods of proving identities.

MATHEMATICS HIP-HOP
A-Level Pure Mathematics | Chapter 04 — Trigonometry | Part 1
Learn it. Practice it. Master it. Share it.