22Etrigfunctions

toc = Trigonometric Functions =

Graphs of Sine and Cosine
The sine curve has many applications in physics. For example, sound, light and electromagnetic waves all travel in a sine curve. Waves travelling through water out on open water travel in an approximate sine curve (affected by friction and air resistance).

For all trig graphs
 * the ** median line ** is the horizontal line through the centre of the graph
 * the ** amplitude ** is the maximum height above the median line
 * the ** period ** is the distance in the x-direction to complete one cycle of the graph

Graph of y = sin(x)
The graph of y = sin(x) {sometimes called the sine wave} is derived from the definition of sine.
 * sin(x) is the y-coordinate of the point on the unit circle created by the angle x.




 * Note: ** The above diagram shows x values in radians. 2p radians equals 360º.

The above diagram shows only one cycle of sine. The sine curve continues to infinity. For the graph y = sin(x)  The x-inctercepts can be found by solving sin(x) = 0
 * the median line is along the x-axis (y = 0)
 * the amplitude is 1 (maximum height above median line)
 * the period is 360º (distance in x-direction to complete one full cycle)
 * in radians, the period is 2p
 * x-intercepts = ..., –180º, 0º, 180º, 360º, 540º, ...

Graph of y = cos(x)
The graph of y = cos(x) is the same shape as y = sin(x) but shifted 90º to the left.

For the graph y = cos(x)
 * the median line is along the x-axis (y = 0)
 * the amplitude is 1 (maximum height above median line)
 * the period is 360º (distance in x-direction to complete one full cycle)
 * in radians, the period is 2p
 * notice the first maximum is at 0º in cos(x) compared to at 90º in sin(x)

The x-inctercepts can be found by solving cos(x) = 0
 * x-intercepts = ..., –90º, 90º, 270º, 450º, ...

Dilations in the Y-Direction

 * y = a ** . ** sin(x), ** ... ** y = a ** . ** cos(x) ** ... ... ... where **a** is any real number

Multiplying sin(x) or cos(x) by ** a ** causes the graph to ** dilate ** (stretch or compress) in the y-direction by a factor of ** a **.

Because the original __**amplitude**__ of sin(x) and cos(x) is ** 1 **, multiplying by a factor of ** a ** changes the amplitude to ** a **.

If ** a ** is __**negative**__, we get a ** reflection ** across the x-axis (in the y-direction).


 * Note: ** Since __**amplitude**__ has to be positive, we take amplitude as the absolute value of a ** (amplitude = |a|) **


 * Example 1a **

Sketch y = 2sin(x)

math . \qquad \bullet \quad \textbf{Amplitude } = \big| 2 \big| = 2 math



For the graph y = 2sin(x)
 * the median line is along the x-axis (y = 0)
 * the amplitude is 2
 * the period is 360º


 * Example 1b **

Sketch y = –0.5cos(q )

math . \qquad \bullet \quad \textbf{Amplitude } = \big| -0.5 \big| = 0.5 math

math . \qquad \bullet \quad \text{Reflected across the x-axis} math

math . \qquad \bullet \quad \cos \big( 0 \big) = 1 \quad \textit{so} \quad -0.5\cos \big(0\big) = -0.5 math



For the graph y = –0.5cos(q )
 * the median line is along the x-axis (y = 0)
 * the amplitude is 0.5
 * the period is 360º

Dilations in the X-Direction

 * y = sin(nx), ** ... ** y = cos(nx) ** ... ... ... where **n** is any real number

Multiplying the x inside sin(x) or cos(x) by ** n ** causes the graph to ** dilate ** (stretch or compress) math \text{in the x-direction by a factor of } \dfrac{1}{n} math

Because the original __**period**__ of sin(x) and cos(x) is ** 360º (or 2 p ) **, multiplying by a factor math \text{of } \dfrac{1}{n} \text{ changes the period to } \dfrac{360^\circ}{n} \; \left( \text{ or } \dfrac{2\pi}{n} \right) math

If ** n ** is __**negative**__, we get a ** reflection ** across the y-axis (in the x-direction).


 * Note: ** Since __**period**__ has to be positive, we take period as the absolute value of 360 ÷ n

math . \qquad \bullet \quad \textbf{Period } = \dfrac{360^\circ}{\big| n \big| } \; \left( \text{ or } \dfrac{2\pi}{\big| n \big| } \right) math


 * Example 2a **

Sketch y = cos(2x)

math . \qquad \bullet \quad \textbf{Period } = \dfrac{360^\circ}{\big| 2 \big| } = 180^\circ math

For the graph y = cos(2x)
 * the median line is along the x-axis (y = 0)
 * the amplitude is 1
 * the period is 180º
 * in radians, the period is p


 * Example 2b **

Sketch y = sin(–0.5q )

math . \qquad \bullet \quad \textbf{Period } = \dfrac{360^\circ}{\big| -0.5 \big| } = 720^\circ math

math . \qquad \bullet \quad \text{Reflected across y-axis} math

math . \qquad \bullet \quad \sin \big( -0.5 \times 180^\circ \big) = \sin \big( -90^\circ \big) = -1 \qquad \big( 180^\circ, \; -1\big) math



Summary

 * y = a ** . ** sin(nx), ** ... ** y = a ** . ** cos(nx) ** ... ... ... where **a,** **n** are any real numbers

math \\ . \qquad \bullet \quad \text{Median line } y = 0 \\. \\ . \qquad \bullet \quad \text{Amplitude } = \big| a \big| \\. \\ . \qquad \bullet \quad \text{Period } = \dfrac{360^\circ}{\big| n \big| } \; \left( \text{ or } \dfrac{2\pi}{ \big| n \big| } \right)\\. \\ . \qquad \bullet \quad \text{If } a < 0 \text{ then reflect across x-axis} \\. \\ . \qquad \bullet \quad \text{If } n < 0 \text{ then reflect across y-axis} math

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