Trigonometric Graphs

Trigonometric Graph

Lesson Objectives

Understand the general shapes of sine, cosine, and tangent graphs.
Identify key characteristics such as amplitude, period, domain, and range.
Interpret the effects of transformations like phase shifts and vertical scaling.
Sketch and analyze the basic and transformed graphs of trigonometric functions.

Lesson Introduction

This lesson introduces the graphs of the three primary trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Understanding these graphs is essential in modeling periodic phenomena such as sound waves, light waves, and the motion of pendulums. The lesson explores how changes in amplitude, period, and phase shift affect the graphs.

Lesson Content

This lesson introduces the graphs of the basic trigonometric functions: sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)).

Graph of the Sine Function

The sine function is written as:

\[ y = \sin x \]

Amplitude: 1
Period: \(360^\circ\) or \(2\pi\) radians
Domain: \( -\infty < x < \infty \)
Range: \( -1 \leq y \leq 1 \)

Example 1: Plot \( y = \sin x \) from \( 0^\circ \) to \( 360^\circ \)

Find sine values at \( 0^\circ, 90^\circ, 180^\circ, 270^\circ, 360^\circ \)
Plot: \((0, 0), (90, 1), (180, 0), (270, -1), (360, 0)\)

Example 2: Plot \( y = 2\sin x \)

Increased amplitude → stretches vertically.

Key points: \((0,0), (90,2), (180,0), (270,-2), (360,0)\)

Graph of the Cosine Function

\[ y = \cos x \]

Amplitude: 1
Period: \(360^\circ\)
Domain: \( -\infty < x < \infty \)
Range: \( -1 \leq y \leq 1 \)

Example 1: Plot \( y = \cos x \)

Points: \((0,1), (90,0), (180,-1), (270,0), (360,1)\)

Example 2: Plot \( y = \cos(x - 90^\circ) \)

This is a horizontal shift to the right by \(90^\circ\), turning cosine into sine.

Graph of the Tangent Function

\[ y = \tan x \]

Period: \(180^\circ\)
Vertical asymptotes at \( x = 90^\circ, 270^\circ, \ldots \)
Domain: \( x \ne 90^\circ + 180^\circ n \)
Range: \( -\infty < y < \infty \)

Example 1: Plot \( y = \tan x \) from \( -180^\circ \) to \( 180^\circ \)

Vertical asymptotes at \( x = -90^\circ, 90^\circ \)
Key points: \((-180, 0), (-135, 1), (0, 0), (45, 1), (180, 0)\)

Example 2: Plot \( y = \tan 2x \)

Halves the period to \(90^\circ\), so the graph repeats more frequently.

Exercise

[WAEC] Sketch the graph of \( y = 2\sin x \) for \( 0^\circ \leq x \leq 360^\circ \). State amplitude and period. [Past Question]
Draw \( y = \cos(x + 60^\circ) \), \( 0^\circ \leq x \leq 360^\circ \). Describe the transformation.
[WASSCE] Sketch \( y = \tan x \) from \( -180^\circ \) to \( 180^\circ \), indicating asymptotes. [Past Question]
Compare \( y = \sin x \) and \( y = -\sin x \) over one period. Comment on symmetry.
[NECO] Sketch \( y = 3\cos x \) for \( 0^\circ \leq x \leq 360^\circ \). State amplitude and minimum value. [Past Question]
Plot \( y = \sin(x - 90^\circ) \). Identify phase shift and compare to the parent function.
Draw \( y = \tan 2x \) for \( -90^\circ \leq x \leq 90^\circ \). What is the new period?
[WAEC] Sketch \( y = \cos 2x \) for \( 0^\circ \leq x \leq 360^\circ \). Label all intercepts. [Past Question]
Solve \( \sin x = \cos x \) for \( 0^\circ \leq x \leq 360^\circ \) and verify graphically.
Graph \( y = -2\sin(x + 45^\circ) \). Identify amplitude, period, and phase shift.