6.2 The Graphs of Sine, Cosine and Tangent Functions
Trigonometric functions such as sine, cosine, and tangent produce periodic curves that model many repetitive real-life phenomena, such as electrocardiogram (ECG) heart readings, respiratory cycles, and tidal movements. This topic explores the basic shapes and properties of \(y = \sin x\), \(y = \cos x\), and \(y = \tan x\) for \(0^\circ \leqslant x \leqslant 360^\circ\), investigates how constants \(a\), \(b\), and \(c\) modify their amplitude, period, and vertical position, and demonstrates how these trigonometric graphs are applied to solve real-world problems.
Learning Objectives
- Draw and identify graphs of trigonometric functions \(y = \sin x\), \(y = \cos x\), and \(y = \tan x\) for \(0^\circ \leqslant x \leqslant 360^\circ\), and compare their characteristics.
- Investigate and make generalisations about the effects of changes in constants \(a\), \(b\), and \(c\) on the graphs of \(y = a \sin bx + c\), \(y = a \cos bx + c\), and \(y = a \tan bx + c\) for \(a > 0, b > 0\).
- Determine the amplitude and period of given trigonometric functions.
- Sketch trigonometric functions with various values of \(a\), \(b\), and \(c\).
- Solve problems involving graphs of sine, cosine, and tangent functions in practical contexts.
Comparison of Basic Trigonometric Graphs and Effects of Constants \(a, b, c\)
Identify sine, cosine, and tangent graphs, and recognise how \(a\) alters amplitude, \(b\) alters frequency/period, and \(c\) causes vertical translation.
Minimum: \(-1\) at \(270^\circ\)
\(x\)-intercepts: \(0^\circ, 180^\circ, 360^\circ\)
Period: \(360^\circ\)
Amplitude: \(1\)
Minimum: \(-1\) at \(180^\circ\)
\(x\)-intercepts: \(90^\circ, 270^\circ\)
Period: \(360^\circ\)
Amplitude: \(1\)
Minimum: \(-\infty\)
Asymptotes: \(90^\circ, 270^\circ\)
Period: \(180^\circ\)
No amplitude
\(\text{Period for sine/cosine} = \dfrac{360^\circ}{b}\)
\(\text{Period for tangent} = \dfrac{180^\circ}{b}\)
Brief Explanation
What is a trigonometric graph?
A trigonometric graph displays the relationship between an angle \(x\) (usually in degrees) and its trigonometric ratio \(y\). The period is the interval of one complete cycle.
What is amplitude?
The amplitude is the maximum distance measured from the centre line to the crest or trough: \(\text{Amplitude} = \dfrac{\text{maximum value} - \text{minimum value}}{2}\).
| Effect | \(y = a \sin bx + c\) | \(y = a \cos bx + c\) | \(y = a \tan bx + c\) |
|---|---|---|---|
| Change in \(a\) | Maximum and minimum values change. Amplitude \(= a\). | Maximum and minimum values change. Amplitude \(= a\). | Graph curvature changes. Has no amplitude. |
| Change in \(b\) | Period \(= \dfrac{360^\circ}{b}\). As \(b\) increases, the graph compresses horizontally. | Period \(= \dfrac{360^\circ}{b}\). As \(b\) increases, the graph compresses horizontally. | Period \(= \dfrac{180^\circ}{b}\). As \(b\) increases, the graph compresses horizontally. |
| Change in \(c\) | Vertical translation. If \(c > 0\), shifts up. If \(c < 0\), shifts down. | Vertical translation. If \(c > 0\), shifts up. If \(c < 0\), shifts down. | Vertical translation. If \(c > 0\), shifts up. If \(c < 0\), shifts down. |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
Determine the amplitude and the period of the function \(y = 4 \cos 2x\).
Why do the values of \(\tan 90^\circ\) and \(\tan 270^\circ\) become undefined on the tangent graph?
State the maximum value and the minimum value of the function \(y = 2 \cos x + 1\).
Common Mistakes
- Confusing the period formula of tangent with sine/cosine: using \(\dfrac{360^\circ}{b}\) for tangent instead of the correct formula \(\text{Period} = \dfrac{180^\circ}{b}\).
- Stating an amplitude value for tangent graphs: tangent graphs extend to \(\pm\infty\) and have no amplitude.
- Omitting the vertical shift \(c\) when calculating maximum and minimum values, for example assuming \(y = a \cos bx + c\) has a maximum of just \(a\) instead of \(a + c\).
- Misidentifying the \(y\)-intercepts: forgetting that \(y = \sin x\) starts at \(0\), whereas \(y = \cos x\) starts at \(1\).
Misconceptions
All periodic functions must have an amplitude.
Tangent is a periodic function repeating every \(180^\circ\), but it has no finite maximum or minimum, so it has no amplitude.
A larger value of \(b\) makes the wave wider.
A larger value of \(b\) increases the number of cycles within \(360^\circ\), compressing the wave horizontally and making the period shorter.
The centre line is always the \(x\)-axis, \(y = 0\).
The centre line shifts vertically with constant \(c\); the centre line is the line \(y = c\).
Summary
- \(y = \sin x\): range \([-1, 1]\), period \(360^\circ\), amplitude \(1\), \(x\)-intercepts at \(0^\circ, 180^\circ, 360^\circ\).
- \(y = \cos x\): range \([-1, 1]\), period \(360^\circ\), amplitude \(1\), \(x\)-intercepts at \(90^\circ, 270^\circ\).
- \(y = \tan x\): range \((-\infty, \infty)\), period \(180^\circ\), no amplitude, undefined at \(x = 90^\circ, 270^\circ\).
- In \(y = a \sin bx + c\) and \(y = a \cos bx + c\): \(a\) sets amplitude, \(b\) determines period \(\dfrac{360^\circ}{b}\), and \(c\) shifts the graph vertically.
- In \(y = a \tan bx + c\): \(a\) changes curvature, \(b\) determines period \(\dfrac{180^\circ}{b}\), and \(c\) shifts the graph vertically.
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 What is the period of the trigonometric function \(y = 3 \tan 2x + 2\)?
2 What are the maximum and minimum values of the function \(y = 3 \cos 3x + 1\)?
B. Fill in the Blanks
3 The maximum distance measured from the centre line to the peak of a sine or cosine wave is known as the .
4 For the function \(y = \cos 4x\), the period of the graph is degrees.
C. Drag and Drop: Match Each Function or Constant to Its Graph Characteristic
Drag each component to the correct characteristic. On a phone, tap the answer choice first, then tap the matching box.
