Trigonometry

2026 Syllabus Objectives

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

  1. Sketch and use graphs of the sine, cosine and tangent functions (for angles of any size, using either degrees or radians)
  2. Use the exact values of sine, cosine and tangent of 30°, 45°, 60°, and related angles
  3. Use the notations sin⁻¹x, cos⁻¹x, tan⁻¹x to denote the principal values of the inverse trigonometric relations
  4. Use the identities sin θ / cos θ = tan θ and sin²θ + cos²θ = 1
  5. Find all the solutions of simple trigonometric equations lying in a specified interval

1. Graphs of Trigonometric Functions

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Understanding the Basic Shapes

Trigonometric functions create wave-like patterns when we plot them on a graph. Let's look at the three main functions: sine, cosine, and tangent.

The Sine Function: y = sin x

The sine function creates a smooth wave that:

  • Starts at 0 when x = 0°
  • Reaches a maximum of 1 at x = 90°
  • Returns to 0 at x = 180°
  • Reaches a minimum of -1 at x = 270°
  • Returns to 0 at x = 360°

This pattern then repeats every 360° (we call this the period of the function).

Key values for the sine grid:

  • At 0°: sin 0° = 0
  • At 90°: sin 90° = 1
  • At 180°: sin 180° = 0
  • At 270°: sin 270° = -1
  • At 360°: sin 360° = 0

The Cosine Function: y = cos x

The cosine function also creates a smooth wave, but it starts differently:

  • Starts at 1 when x = 0°
  • Reaches 0 at x = 90°
  • Reaches a minimum of -1 at x = 180°
  • Returns to 0 at x = 270°
  • Returns to 1 at x = 360°

Key values for the cosine grid:

  • At 0°: cos 0° = 1
  • At 90°: cos 90° = 0
  • At 180°: cos 180° = -1
  • At 270°: cos 270° = 0
  • At 360°: cos 360° = 1

Important note: The cosine graph looks exactly like the sine graph, but shifted 90° to the left.

The Tangent Function: y = tan x

The tangent function is different from sine and cosine:

  • It has vertical asymptotes (lines the graph approaches but never touches) at x = 90°, 270°, etc.
  • It passes through 0 at x = 0°, 180°, 360°
  • It goes to positive infinity as it approaches 90° from the left
  • It goes to negative infinity as it approaches 90° from the right
  • The pattern repeats every 180° (not 360° like sine and cosine)

Key values for the tangent grid:

  • At 0°: tan 0° = 0
  • At 45°: tan 45° = 1
  • At 90°: tan 90° = undefined (∞)
  • At 135°: tan 135° = -1
  • At 180°: tan 180° = 0

Using Radians Instead of Degrees

Radians are another way to measure angles. Instead of dividing a full circle into 360 degrees, we use the mathematical constant π (pi).

Key conversions:

  • 180° = π radians
  • 90° = π/2 radians
  • 60° = π/3 radians
  • 45° = π/4 radians
  • 30° = π/6 radians
  • 360° = 2π radians

All the graphs work the same way whether you use degrees or radians—you just label the x-axis differently.

Transformations of Trigonometric Graphs

We can transform the basic sine, cosine, and tangent graphs using the general equation:

y = a sin(bx) + c (or cos or tan)

Where:

  • a = amplitude (how tall the wave is)
  • b = frequency (how many complete waves fit in 360°)
  • c = base line (how far up or down the whole graph is shifted)

Amplitude (a)

The amplitude is the distance from the base line to the highest (or lowest) point of the wave.

Example: y = 3 sin x

  • The amplitude is 3
  • The wave reaches 3 at its highest and -3 at its lowest
  • The basic sin grid values (0, 1, 0, -1, 0) become (0, 3, 0, -3, 0)

Example: y = 2 cos x

  • The amplitude is 2
  • The wave reaches 2 at its highest and -2 at its lowest

Base Line (c)

The base line is the horizontal line that the wave oscillates (moves up and down) around. Adding or subtracting a number shifts the entire graph vertically.

Important: The grid values tell you the distance from the base line, not the actual y-values.

Example: y = sin x + 2

  • The base line is at y = 2
  • The wave goes from y = 1 (base line - 1) to y = 3 (base line + 1)
  • Grid values (0, 1, 0, -1, 0) from the base line of 2 give actual y-values of (2, 3, 2, 1, 2)

Example: y = cos x - 1

  • The base line is at y = -1
  • The wave goes from y = -2 to y = 0

Frequency (b)

The frequency tells you how many complete cycles (full waves) occur in 360° (or 2π radians).

Example: y = sin 2x

  • Frequency is 2
  • There are 2 complete waves between 0° and 360°
  • One complete wave takes 180° instead of 360°

Example: y = cos 3x

  • Frequency is 3
  • There are 3 complete waves between 0° and 360°
  • One complete wave takes 120°

For tangent: Remember that tan x repeats every 180°, so frequency works differently.

Example: y = tan(½x)

  • Frequency is ½
  • One complete cycle takes 360° instead of 180°

Combined Transformations

Example: y = 3 sin 2x - 1

Step-by-step approach:

  1. Base line: y = -1 (shift everything down by 1)
  2. Frequency: 2 (two complete waves in 360°)
  3. Apply the grid: sin values at 0°, 90°, 180°, 270°, 360° for ONE wave, but we need TWO waves
    • First wave: 0° to 180° with grid values (0, 1, 0, -1, 0) at 0°, 45°, 90°, 135°, 180°
    • Second wave: 180° to 360° with grid values (0, 1, 0, -1, 0) at 180°, 225°, 270°, 315°, 360°
  4. Amplitude: 3 (multiply all grid values by 3, then add to base line)
    • From base line -1: points are at y = -1, 2, -1, -4, -1

Example: y = 1 - cos 2x

This can be rewritten as: y = -cos 2x + 1

  • Base line: y = 1
  • Frequency: 2
  • Amplitude: -1 (negative means the graph is reflected—flipped upside down)

Squared Trigonometric Functions

When we square a trigonometric function, all negative values become positive.

Example: y = sin²x

The basic sin grid (0, 1, 0, -1, 0) becomes (0, 1, 0, 1, 0) when squared.

  • The graph never goes below the x-axis
  • It oscillates between 0 and 1
  • It completes TWO bumps in 360° (not one wave like regular sin x)

Example: y = 3 cos²x - 2

  1. Square the cos values: grid becomes (1, 0, 1, 0, 1)
  2. Multiply by 3: grid becomes (3, 0, 3, 0, 3)
  3. Shift down by 2: grid values from base line -2 give (1, -2, 1, -2, 1)

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