]> Trigonometric functions

### Chapter 1: Differentiation; Section 2: Real Functions; page 3

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# Trigonometric Functions

## Definitions

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  2π  are divided in  4  quadrants. The graphical presentation of any other angle =  α±2 , where  n  is a natural number, shares the same position in the quadrant as the angle  α .

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)
 $\mathrm{tan}\alpha =\frac{\mathrm{sin}\alpha }{\mathrm{cos}\alpha }$ (1.2.3.2)

We also found that

 $\begin{array}{l}\mathrm{sin}0=\mathrm{cos}\frac{\pi }{2}=0\\ \mathrm{sin}\frac{\pi }{2}=\mathrm{cos}0=1\end{array}\right\}$ (1.2.3.3)

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  2π  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 :

 $\begin{array}{l}\mathrm{sin}\left(\alpha ±\beta \right)=\mathrm{sin}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{cos}\beta ±\mathrm{cos}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{ }\beta \\ \mathrm{cos}\left(\alpha ±\beta \right)=\mathrm{cos}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{cos}\text{ }\beta \mp \mathrm{sin}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{sin}\text{ }\beta \\ \mathrm{tan}\left(\alpha ±\beta \right)=\frac{\mathrm{tan}\text{ }\alpha ±\mathrm{tan}\text{ }\beta }{1\mp \mathrm{tan}\text{ }\alpha \text{\hspace{0.17em}}\mathrm{tan}\text{ }\beta }\end{array}\right\}$ (1.2.3.4)

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)

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:

 $\begin{array}{l}\mathrm{sec}\alpha =\frac{1}{\mathrm{cos}\alpha }\\ \mathrm{csc}\alpha =\frac{1}{\mathrm{sin}\alpha }\\ \mathrm{cot}\alpha =\frac{1}{\mathrm{tan}\alpha }\end{array}\right\}$ (1.2.3.6)

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)

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)
where the variable t is the time, T is the time period of the motion and l the amplitude of the oscilator, both the cosine and the argument of cosine are dimensionless.

The function y=sinx is constructed geometrically and plotted in Fig. Sine.

From the plot of the function y=sinx and by use of (1.2.3.5) one obtains the graphic presentation of y=cosx as given in Fig. Cosine.

The functions y=tanx and y=cotx 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

 $\mathrm{tan}\text{ }\theta =\frac{2t}{1-{t}^{2}}$ (1.2.3.9)
where
 $t=\mathrm{tan}\frac{\theta }{2}$ (1.2.3.10)

From the other relations of (1.2.3.4) and doing the same substitution, one obtains:

 $\begin{array}{l}\mathrm{sin}\text{ }\theta =\frac{2t}{1+{t}^{2}}\\ \mathrm{cos}\text{ }\theta =\frac{1-{t}^{2}}{1+{t}^{2}}\end{array}\right\}$ (1.2.3.11)
where t is from (1.2.3.10). The proof of (1.2.3.11) is given as exercise #2 for the user.

## Inverse trigonometric functions

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.  asinx  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  2π  (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.

The arc-sine function is obtained and discussed in Fig. Arc-sine. We found two branches of arc-sine that give single valued angle, each one in different pair of quadrants:
• 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.

The single valued branches of arc-cosine are in different 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

Each branch of the arc-tangent function is increasing and  $-\frac{\pi }{2}\le \text{atan}\text{ }x\le \frac{\pi }{2}$  is used as reference.
Each branch of the arc-cotangent function is decreasing and  $0\le \text{acot}\text{ }x\le \pi$  is used as reference.

## Exercises

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 cotβ=0.75 and cosβ<0 .

1. In what quadrant is the angle β ?
2. Calculate the values of all the triginometric functions for this β !
3. 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.
1. What is the condition for not having a solution?
2. Find the condition for a unique solution! What is t in this case?
3. Calculate numerically θ in the interval [0,2π) for the case a=b=c !

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