6.3 |
Graph of Sine, Cosine and Tangent Functions |
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Graphs of Trigonometric Functions |
\(y=\sin{x}\) and \(y=\cos{x}\) |
The graphs of \(y=\sin{x}\) and \(y=\cos{x}\) are sinusoidal and have the following properties:
- The maximum value is \(1\) while the minimum value is \(-1\), so the amplitude of the graph is \(1\) unit.
- The graph repeats itself every \(360^\circ\) or \(2\pi\) rad, so \(360^\circ\) or \(2\pi\) rad is the period for both graphs.
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The graph \(y=\tan{x}\) is not sinusoidal. The properties of \(y=\tan{x}\) are as follows:
- This graph has no maximum or minimum value.
- The graph repeats itself every \(180^\circ\) or \(\pi\) rad interval, so the period of a tangent graph is \(180^\circ\) or \(\pi\) rad.
- The function \(y=\tan{x}\) is not defined at \(x=90^\circ\) and \(x=270^\circ\). The curve approaches the line \(x=90^\circ\) and \(x=270^\circ\) but does not touch the line. This line is called an asymptote.
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Graph \(y=\sin{x}\) for \(-2\pi \leq x \leq 2\pi\) |
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- Amplitude \(=1\).
- The maximum value of \(y=1\).
- The minimum value of \(y=-1\).
- Period \(=360^\circ\) or \(2\pi\).
- \(x\)-intercepts: \(-2\pi\), \(-\pi\), \(0\), \(\pi\), \(2\pi\).
- \(y\)-intercepts: \(0\).
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Graph \(y=\cos{x}\) for \(-2\pi \leq x \leq 2\pi\) |
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- Amplitude \(=1\).
- The maximum value of \(y=1\).
- The minimum value of \(y=-1\).
- Period \(=360^\circ\) or \(2\pi\).
- \(x\)-intercepts: \(-\dfrac{3}{2}\pi\), \(-\dfrac{1}{2}\pi\), \(\dfrac{1}{2}\pi\), \(\dfrac{3}{2}\pi\).
- \(y\)-intercepts: \(1\).
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Graph \(y=\tan{x}\) for \(-2\pi \leq x \leq 2\pi\) |
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- No amplitude.
- There is no maximum value of \(y\).
- There is no minimum value of \(y\).
- Period \(=180^\circ\) or \(\pi\).
- \(x\)-asymptotes: \(-\dfrac{3}{2}\pi\), \(-\dfrac{1}{2}\pi\), \(\dfrac{1}{2}\pi\), \(\dfrac{3}{2}\pi\).
- \(x\)-intercepts: \(-2\pi\), \(-\pi\), \(0\), \(\pi\), \(2\pi\).
- \(y\)-intercepts: \(0\).
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Effect of Values \(a\), \(b\) and \(c\) in the Function \(y=a\sin{bx}+c\) |
- The values of \(a\), \(b\) and \(c\) in the function \(y=a\sin{bx}+c\) affect the amplitude, the period and the position of the graph.
- The effects of changing the values of \(a\), \(b\) and \(c\) on the graph can be summarised as follows:
Change in |
Effects |
\(a\) |
The maximum and minimum values of the graphs (except for the graph of \(y=\tan{x}\) where there is no maximum or minimum value). |
\(b\) |
Number of cycles in the range \(0^\circ \leq x \leq 360^\circ\) or \(0 \leq x \leq 2\pi\):
- Graphs \(y=\sin{x}\) and \(y=\cos{x}\) (period \(=\dfrac{360^\circ}{b}\) or \(\dfrac{2}{b}\pi\)).
- Graph \(y=\tan{x}\) (period \(=\dfrac{180^\circ}{b}\) or \(\dfrac{1}{b}\pi\)).
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\(c\) |
The position of the graph with reference to the \(x\)-axis as compared to the position of the basic graph. |
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Example |
State the cosine function represented by the graph above.
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Note that the amplitude is \(4\).
So, \(a=4.\)
Two cycles in the range of \(0^{\circ} \leqslant x \leqslant 2\pi\).
The period is \(\pi\), that is, \(\dfrac{2\pi}{b} = \pi\), so \(b=2\).
Hence, the graph represents \(y= 4\text{ cos }2x\).
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