Trigonometric Methods

2

Trigonometric Methods

 

CONTENTS

2.1       Sinusoidal functions

2.1.2    Review of the trigonometric ratios

2.1.3    Cartesian and polar co-ordinate systems

2.1.4    Properties of the circle

2.1.5    Radian measure

2.1.6    Sinusoidal functions

2.2       Applications

2.2.1    Angular velocity

2.2.2    Angular acceleration

2.2.3    Centripetal force

2.2.4    Frequency , amplitude,  phase

2.2.5        The production of complex waveforms using sinusoidal graphical synthesis

2.2.6    AC waveforms and phase shift

2.3      Trigonometric identities

2.3.1        Relationship between trigonometric and hyperbolic identities

2.3.2        Double angle and compound angle formulae and the conversion of products to sums and differences

2.3.3        Use of trigonometric identities to solve trigonometric equations and simplify trigonometric expressions

2.1Sinusoidal functions

Why is the study of Sinusoidal Waveforms important?

Sinusoidal waveforms are of special interest for a number of reasons:

n  it is a natural form occurring in an oscillator circuit; also the form of voltage induced in a turn (coil) of wire rotated in a magnetic field, ie. a generator

n  it is the form of voltage used for both distribution of electricity and for communications

n  all periodic waveforms can be represented as a series of sine waves using Fourier analysis.

Example: Coil rotating in a magnetic field. Induced voltage and resulting current in a coil
rotating in a magnetic field is sinusoidal. Fig 2.1 shows how the brushes are arranged so that the voltage induced between the two terminals of a rotating coil in a magnetic field is tapped. Arrangement in Fig 2.1 (top) gives an alternating current (AC) output and Fig 2.1 (bottom) shows how a direct current (DC) output is obtained.

Fig. 2.1 Induced voltage and resulting current in a coil rotating in a magnetic field is sinusoidal

We can also express a sinusoidal function using cosine function rather than the sine function itself. The functions are related by the identity

 

Sinusoidal Waveform, v(t) =                         :

Fig 2.2 shows the waveform. Following definitions are important.

V_m is the peak value

ω is the angular frequency in radians per second

θ is the phase angle

T is the period

 

Fig. 2.2 Sinusoidal waveform, v(t) =

Review of the trigonometric ratios:

c

b

a

C

B

A

Following figure shows a right-angle triangle ABC. Recall the trigon ometrc ratios

                                                                                                  

                                                                                           

                                                                                                  

                                                                                         

Hence we have

From Pythagora’s theorem,  Therefore, we have the following trigonometric identity

Note that this identity holds for any angle, A. Following trigonometric identities are mentioned without proof. Note that these identities are true for angles of any size.

These trigonometric identities can be used to solve trigonometric equations and simplify trigonometric expressions.

Use the above trigonometric identities to show the results in the following table.