Trigonometric Functions


Trig functions are invaluable for applications and provide important examples. They link Calculus to Geometry. In Calculus, all trigonometric functions are functions of radians

Standard Notation

The functions sin(x) and cos(x) are defined by the picture on the right.

The other trigonometric functions are defined by

  • tan(x) = sin(x)/cos(x)
  • cot(x) = cos(x)/sin(x)
  • sec(x) = 1/cos(x)
  • csc(x) = 1/sin(x)

When expressing positive integer powers of trig functions, we write the exponent directly after the name of the function. Thus cos2(x) means [cos(x)]2. The notation cos-1(x) is reserved for the inverse cosine which is also called "arccosine" and can be written as arccos(x) or, on many calculators, acos(x). The same applies to inverse sine, inverse tangent, and so on.


From the Pythagorean relation on the right triangle OPQ, it is clear that

cos2(x) + sin2(x) = 1.

This important relation is called an identity. Identities are equations which are true for all values of the variable. Some other useful identities are

  • sin(x + y) = sin(x) cos(y) + sin(y) cos(x)
  • cos(x + y) = cos(x) cos(y) - sin(x) sin(y)
  • sin(2x) = 2 sin(x) cos(x)
  • cos(2x) = cos2x - sin2x
  • 1 + tan2(x) = sec2(x)
  • 1 + cot2(x) = csc2(x)

AND THERE ARE LOTS MORE which we will not list here.

Graphs of Sin and Cosine

The most commonly used trigonometric functions used in calculus are sin(x), cos(x) and tan(x). We'll leave it to you to review any information you need on the other three functions. The graphs of sin (x) and cos (x) have several distinct features.

Notice that both graphs repeat themselves over an interval of 2.

The sine and cosine functions are said to be periodic with period 2:

sin (x + 2n) = sin (x) and cos (x + 2n) = cos x for any integer n.

The maximum value attained by sin(x) or by cos (x) is 1, and the minimum value is -1. The functions oscillate in a regular manner within 1 unit of the x-axis (y = 0). We say that the amplitude is 1.

Sin (x) is an odd function because sin(-x) = -sin(x). It's graph is symmetric to the origin. Cos (x) on the other hand is an even function cos (-x) = cos (x), and its graph is symmetric to the y-axis.

Notice also that the graphs are exactly the same shape but horizontally shifted /2 units. They are related by: sin (x) = cos (x - /2) and cos (x) = sin (x + /2).

The general form of a sinusoidal function is f(x) = Asin B(x - C) or f(x) = Acos B(x - C), where A, B, and C are real numbers. The amplitude is A, the period is given by (2)/B and the constant C is called the phase shift.

The Tangent Function

Now let's turn our attention to the tangent function.

Since tan(x) = sin(x)/cos(x), its domain is all real numbers except those at which cos (x) = 0. At these points, the odd multiples of (/2), the graph of tan (x) has vertical asymptotes. Tan (x) has zeros where sin (x) = 0, that is at multiples of . The graph at right shows that tan (x), too, is periodic, but unlike sin (x) and cos (x), it's period is .

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