Mathematical Introduction for Physics and Engineering

by Samuel Dagan (Copyright © 2007-2020)

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The variable of a trigonometric function is a **plane angle**. The definitions of an angle is presented on **Fig. Angles** .

Since the angle is defined by the ratio of two lengths, it does not have physical dimensions. As shown, the **units** of measuring an angle used here **are called radians. π radians correspond to 180°** .

The angles *α* between 0 and 2*π* are divided in 4 quadrants. The graphical presentation of any other angle = ** α±2nπ** , where

The basic trigonimetric functions **sin (sine), cos (cosine) and tan (tangent)** are defined graphically in **Fig. Trigonometric functions** and the following relations are given:

${\mathrm{sin}}^{2}\alpha +{\mathrm{cos}}^{2}\alpha =1$ | (1.2.3.1) |
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$\mathrm{tan}\alpha =\frac{\mathrm{sin}\alpha}{\mathrm{cos}\alpha}$ | (1.2.3.2) |
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We also found that

$\begin{array}{l}\mathrm{sin}0=\mathrm{cos}\frac{\pi}{2}=0\\ \mathrm{sin}\frac{\pi}{2}=\mathrm{cos}0=1\end{array}\}$ | (1.2.3.3) |
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From the definition of **the functions** it is clear that they **are periodic**, meaning that they repeat themselves over and over again after an interval of the variable, called period. **The period is 2 π for sin and cos and π for tan** . This can be seen again from the graphic presentations later on.

The following table summarizing the signs of the trigonometric functions by quadrants is very useful. It yields that **the sign of any two of the functions** sine, cosine and tangent **defines the quadrant** of the variable.

Quadrants: | Q1 | Q2 | Q3 | Q4 |

sine | + | + | − | − |

cosine | + | − | − | + |

tangent | + | − | + | − |

It can be shown that for any *α* and *β* :

$\begin{array}{l}\mathrm{sin}\left(\alpha \pm \beta \right)=\mathrm{sin}\text{\hspace{0.05em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\beta \pm \mathrm{cos}\text{\hspace{0.05em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.05em}}\beta \\ \mathrm{cos}\left(\alpha \pm \beta \right)=\mathrm{cos}\text{\hspace{0.05em}}\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.05em}}\beta \mp \mathrm{sin}\text{\hspace{0.05em}}\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.05em}}\beta \\ \mathrm{tan}\left(\alpha \pm \beta \right)=\frac{\mathrm{tan}\text{\hspace{0.05em}}\alpha \pm \mathrm{tan}\text{\hspace{0.05em}}\beta}{1\mp \mathrm{tan}\text{\hspace{0.05em}}\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{\hspace{0.05em}}\beta}\end{array}\}$ | (1.2.3.4) |
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These relations are assumed to be familiar from high-school and their proof is not repeated here. The last relation is easily obtained from the first two by use of (1.2.3.2).

By substituting $\beta =\frac{\pi}{2}$ in (1.2.3.4) and in view of (1.2.3.3) one obtains easily that

$\mathrm{cos}\alpha =\mathrm{sin}\left(\alpha +\frac{\pi}{2}\right)$ | (1.2.3.5) |
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which together with (1.2.3.2) yields that **the function cos and tan can be obtained from sin** .

The **reciprocals** of these trigonometric functions are **seldom used**. Their names are : sec (secant), csc (cosecant) and cot (cotangent). Here are their definitions:

$\begin{array}{l}\mathrm{sec}\alpha =\frac{1}{\mathrm{cos}\alpha}\\ \mathrm{csc}\alpha =\frac{1}{\mathrm{sin}\alpha}\\ \mathrm{cot}\alpha =\frac{1}{\mathrm{tan}\alpha}\end{array}\}$ | (1.2.3.6) |
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From (1.2.3.2), (1.2.3.5) and (1.2.3.6) one obtains easily that

$\mathrm{cot}\text{\hspace{0.17em}}\alpha =-\mathrm{tan}\left(\alpha +\frac{\pi}{2}\right)$ | (1.2.3.7) |
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As in the case of angles, **the trigonometric functions** are also physically **dimensionless**. As an example one can see that in the expression of **harmonic motion**

$y=l\mathrm{cos}\left(\frac{2\pi t}{T}\right)$ | (1.2.3.8) |
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The function *y*=sin*x* is constructed geometrically and plotted in **Fig. Sine**.

From the plot of the function *y*=sin*x* and by use of (1.2.3.5) one obtains the graphic presentation of *y*=cos*x* as given in **Fig. Cosine**.

The functions *y*=tan*x* and *y*=cot*x* are plotted and discussed in **Fig. Tangent &
co**.

From the graphic presentation of the **trigonometric functions** it was shown that all of them **are odd** with the **exception of the cosine** (and of course also the secant) which are even.

The **following relations are widely used** in order to transform trigonometric expressions in algebraic form. By substituting *α* and *β* by
$\frac{\theta}{2}$
in (1.2.3.4) one obtains

$\mathrm{tan}\text{\hspace{0.05em}}\theta =\frac{2t}{1-{t}^{2}}$ | (1.2.3.9) |
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$t=\mathrm{tan}\frac{\theta}{2}$ | (1.2.3.10) |
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From the other relations of (1.2.3.4) and doing the same substitution, one obtains:

$\begin{array}{l}\mathrm{sin}\text{\hspace{0.05em}}\theta =\frac{2t}{1+{t}^{2}}\\ \mathrm{cos}\text{\hspace{0.05em}}\theta =\frac{1-{t}^{2}}{1+{t}^{2}}\end{array}\}$ | (1.2.3.11) |
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Each **inverse triginometric function is called arc-function**, e.g. the inverse of sine is called: arc-sine. The name arc emphasizes that the inverse of a trigonometric function expresses an angle, which is actually an arc of a circle with a radius of unity. We are going to use the letter "a" in front of the trigonometric function for denoting its inverse. E.g. **asin x means arc-sine of x** . One can find elsewhere also the use of "arc" as a prefix instead of "a" .

Since the trigonometric functons are periodic, the inververse functions are multiple-valued. However if one limits the angles in an interval of 2*π* (four quadrants) the arc-sine and arc-cosine become double valued. As we'll see since the tangent and cotangent have a period of *π* by a choice of an interval of *π* their inverse functions become single valued. It is common to choose **one branch of the function as reference (called also the main branch)** in order to limit the arc-functions to single-valued.

- Q4 and Q1, e.g. $-\frac{\pi}{2}\le \text{a}\text{}\mathrm{sin}x\le \frac{\pi}{2}$ increasing function - reference
- Q2 and Q3, e.g. $\frac{\pi}{2}\le \text{a}\text{}\mathrm{sin}x\le \frac{3\pi}{2}$ decreasing function

The deduction of the arc-cosine function is similar to that of arc-sine as seen from
**Fig. Arc-cosine**. However there is difference in grouping the quadrants.

- Q1 and Q2, e.g. $0\le \text{a}\text{}\mathrm{cos}x\le \pi $ decreasing function - reference
- Q3 and Q4, e.g. $\pi \le \text{a}\text{}\mathrm{cos}x\le 2\pi $ increasing function

The arc-tangent and arc-cotangent functions are obtained and discussed at **Fig. Arc-tangent & co**

**Exercise 1.** Prove the equations (1.2.3.5) and (1.2.3.7) with the aid of (1.2.3.4)!

**Exercise 2.** Prove the relations (1.2.3.11) with *t* defined at (1.2.3.10) !

**Exercise 3.** Given **cot β=0.75 and cosβ<0** .

- In what quadrant is the angle
*β*? - Calculate the values of all the triginometric functions for this
*β*! - What is
*β*in radians for the interval (−*π*,*π*] ? Use a math table or calculator.

**Exercise 4.** Given: **
$a\mathrm{sec}\theta +b\mathrm{tan}\theta =c$
** , where ** a,b and c are constants≠0**.

Use the variable *t* from (1.2.3.10) together with (1.2.3.9) and (1.2.3.11) for solving *θ*.

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- What is the condition for not having a solution?
- Find the condition for a unique solution! What is
*t*in this case? - Calculate numerically
*θ*in the interval [0,2*π*) for the case*a=b=c*!

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