10.1 Trigonometric Function Definitions

2026 Syllabus Objectives

By the end of these notes, you will be able to:

  • Understand what an angle of any magnitude means (including negative angles and angles greater than 360°)
  • Define and use all six trigonometric functions: sine, cosine, tangent, secant, cosecant, and cotangent
  • Know which functions are positive or negative in each quadrant
  • Apply these definitions to find trigonometric values for any angle

1. What Is an Angle?

https://cambridgeassistant.com

An angle is not just something inside a triangle. In this topic, we think of an angle as a rotation.

Imagine a line, called OP, that starts flat along the positive x-axis (pointing right) and then rotates around the fixed point O (the origin — the centre of the graph).

  • The angle θ (Greek letter "theta") measures how much the line has rotated from the positive x-axis.
  • Anticlockwise (the direction opposite to a clock) rotation → positive angle
  • Clockwise (the same direction as a clock) rotation → negative angle

This means angles can be:

  • Between 0° and 360° (one full turn)
  • Greater than 360° (more than one full turn — the line keeps spinning)
  • Negative (the line spins clockwise instead)

2. The Four Quadrants

https://cambridgeassistant.com

The x-axis (horizontal) and y-axis (vertical) divide the flat plane into four regions called quadrants. We number them like this:

        y
        |
   2nd  |  1st
        |
--------+--------  x
        |
   3rd  |  4th
        |
Quadrantx valuey valueAngle range (°)
1stpositive (+)positive (+)0° – 90°
2ndnegative (−)positive (+)90° – 180°
3rdnegative (−)negative (−)180° – 270°
4thpositive (+)negative (−)270° – 360°

The quadrant an angle is in means: the quadrant where the rotating line OP ends up.

Finding Which Quadrant an Angle Is In

Step 1: If the angle is greater than 360°, subtract 360° until it is between 0° and 360°. Step 2: If the angle is negative, it is a clockwise rotation.

Example: Where does 490° land? 490° − 360° = 130°. Since 90° < 130° < 180°, it is in the 2nd quadrant.

Example: Where does −70° land? A clockwise rotation of 70° from the positive x-axis puts us in the 4th quadrant.

Example: Where does 240° land? Since 180° < 240° < 270°, it is in the 3rd quadrant.

Sign in to view full notes