On-line Math 21

3.1  Trigonometric functions

3.1.1  Basic definitions

This is intended to be a reminder of the basic properties of trigonometric functions. It is too brief to replace a course in trigonometry, and it focuses on only those properties of trigonometric functions that have particular use for calculus. The classical uses of trigonometry, such as surveying and other tasks where the measurement of various lengths and angles are the primary object, are not developed. Also, the many trigonometric identities that comprise a large part of most trigonometry courses are cut back to only those identities that are useful for calculus.

Right triangles, adjacent sides, and so on.

Trigonometric functions are disconcerting to a number of students because they are the first functions they see which are not really described algebraically. They are a different sort of function, whose best description is through geometry. The first definition is for the sine of an angle q, written sinq. In order to even define sinq, you have to first construct a right triangle with angle q, then label the sides opposite to q, adjacent to q, and the hypotenuse.

There are 5 other trigonometric functions, the cosine ( cosq), tangent ( tanq), cotangent ( cotq), secant ( secq), and cosecant ( cscq). They can all be defined similarly to sinq, according to the following table:

sinq = opposite hypotenuse cosq = adjacent hypotenuse tanq = sinq cosq = opposite adjacent cotq = cosq sinq = 1 tanq = adjacent opposite secq = 1 cosq = hypotenuse adjacent cscq = 1 sinq = hypotenuse opposite .

There are several things to point out about these definitions. The first is that they are all ratios of sides of the triangle. The reason for that is that these functions are not supposed to depend on the size of the triangle, only on the size of the angle. From geometry, the ratios of sides are independent of the size of the triangle, as long as the angles are the same. (as long as the triangles are similar). The second thing to notice is that I listed first the expressions of one trigonometric function in terms of others, if possible. Always remember that any trigonometric function can be written in terms of sinq and cosq.

Angles and the unit circle

While the previous paragraph is the first way you probably saw trigonometric functions defined, it was not the best way. For our purposes we need to re-cast the definitions in terms of the unit circle. Most importantly, we need these functions to be functions of a real number, not an angle (or even an angle in degrees).

Imagine an angle being represented by a ray from the origin. That ray makes the angle q with the positive horizontal ( x ) axis. The radian measure of the angle q is defined as the distance from the positive x -axis to the ray along the unit circle, measured counter-clockwise. The unit circle is the circle of points a distance 1 from the origin. That circle is given by the equation x²+y² = 1 .

From the context of the unit circle, the functions cosq and sinq of the angle q in radian measure (or, of the number q) is defined as the x - and y -coordinates of the point where the ray at angle q crosses the unit circle, x = cosq and y = sinq.

sin(x) and cos(x) as functions.

Now we want to separate these trigonometric functions from the unit circle, and think of them as functions. There is a source of confusion in doing this, since we now will deal with sin(x) , where the x refers not to the x -coordinate on the unit circle, but the number corresponding to an angle of radian measure x . If we do have to think about the unit circle explicitly, we will need a new label for the horizontal axis - probably calling the axes the u and v axes makes it easier.

As a function, what does sin(x) look like? Since sin(x) is the horizontal component of a point on the unit circle, then certainly it is a number between -1 and +1 . Since sinq makes since for any angle q, sin(x) is defined for all x . These two observations say that sin(x) has domain the entire real line, and range the interval [-1,1] . Following along the circle above, you can see that the graph of y = sin(x) looks like the following:

On thing you should notice is that the plots seem to be similar. In particular, the graph of cos(x) is nearly the same as that of sin(x) . In fact, they actually are the same, except that one is shifted to one side from the other. The best way to see that shift is to think about the plot of sin(x-a) , which is the function whose value at x is the same as sin(x) has at x-a . So, the graph of sin(x-a) is the same as that of sin(x) but shifted to the right by a units. Try that in the following graphic, comparing the graph of sin(x) to sin(x-a), and to cos(x), as you change a . Find the value of a for which sin(x-a) = cos(x).

[This should be a ``toy'' which plots sin(x), cos(x) and sin(x-a) simultaneously, after the user chooses the value for a ].

This ``shifting'' of the plot by a units to the right by replacing x by (x-a) everywhere in the formula will of course work for all functions, and you can see for example that cot(x) is a shift of tan(x) and csc(x) is a shift of sec(x) .

Trigonometric identities

A trigonometric identity is a relationship between trig functions, such as

cos(x) = sin æ ç è p 2 -x ö ÷ ø ,

which holds for all values of x . Trigonometric identities account for hundreds of pages, hundreds of different identities, in a trigonometry text. We are not going to re-construct that here. We are, instead, going to concentrate on a very few of these identities (some of which will be exercises), the ones which are most important for calculus.

There are a number of easy identities that follow from the definitions of sinx and cosx . For example, since -x is the same angle as x , but wrapped the other way around the unit circle,

sin(-x) = -sin(x), and cos(-x) = cos(x),

which you can see from the drawing.

Where does this fundamental identity come from? Well, since (cos(x),sin(x)) is a point on the unit circle u²+v² = 1 (remember, the axes of the unit circle are u and v now), taking u = cos(x) and v = sin(x) , the equation of the circle is the same as cos²x+sin²x = 1 , which is the same as the identity except that the order of the sum is reversed, which doesn't matter.

This identity is used in countless ways, as we will see. The most common use of it is to replace cosx , say, by

cos(x) = Ö   1-sin2x   .

Why you would want to do that is another question, and will be clear when it is needed. Many other identities are derived from it, such as

tan2x+1 = sec2x,

which is exactly the same identity, divided by cos²x , with the names for the ratios sinx/cosx = tanx and 1/cosx = secx substituted in.

There is one other (yes, really, only one) fundamental identity. All others follow from this one, the ones we have already discussed, and the definitions of the trig functions. That identity is the addition rule - well, it's really two, since there are addition rules for sine and cosine.

sin(A+B) = sin(A)cos(B)+cos(A)sin(B) cos(A+B) = cos(A)cos(B)-sin(A)sin(B)

We'll prove the second of these identities, by looking at a picture: Proof

Using these identities, you can (yes, you can) show that the following long list of identities hold. Some are all set up as examples that you can work through.

cos(A+B) = cos(A)cos(B)-sin(A)sin(B)

sin(A+B) = sin(A)cos(B)+cos(A)sin(B)

cos(A-B) = cos(A)cos(B)+sin(A)sin(B)

sin(A-B) = sin(A)cos(B)-cos(A)sin(B)

tan(A+B) = tan(A)+tan(B) 1-tan(A)tan(B)

tan(A-B) = tan(A)-tan(B) 1+tan(A)tan(B)

cos(2x) = cos2(x)-sin2(x)

sin(2x) = 2sin(x)cos(x)

The identity for cos(2x) generates, along with the fundamental identity cos²(x)+sin²(x) = 1 , two more identities that are then used for some additional identities that are very useful in calculus.

cos(2x) = cos2(x)-sin2(x) = 2cos2(x)-1 = 1-2sin2(x).

Solving for cos²(x) or sin²(x) gives

cos2(x) = 1 2 (1+cos(2x)), sin2(x) = 1 2 (1-cos(2x)).

These lead, as well, to half-angle formulas (substituting t = 2x and taking square roots)

cos(t/2) =   æ  ú Ö 1 2 (1+cos(t))   , sin(t/2) =   æ  ú Ö 1 2 (1-cos(t))   .

Complementary angles and periodicity

The terms ``cosine'', ``cotangent'', and ``cosecant'' refer to the fact that the cosine of an angle q is the same as the sine of the complementary angle, the other angle of a right triangle. That is,

cos(q) = sin æ ç è p 2 -q ö ÷ ø ,

and so on. Those are another form of trigonometric identity, and can be verified by applying the subtraction formulas where A = p/2 and B = q. That is,

cos(q) = sin æ ç è p 2 -q ö ÷ ø cot(q) = tan æ ç è p 2 -q ö ÷ ø csc(q) = sec æ ç è p 2 -q ö ÷ ø .

Similarly, because when you change from an angle with measure q to q+2p, the ray from the origin does not change, any trig function will be the same at q+2p as it was at q, that is,

sin(q+2p) = sin(q).

This reflects the periodicity of trigonometric functions, the fact that, after an interval of 2p, the function repeats its values from the previous interval.

Copyright (c) by David L. Johnson, last modified
On 25 Apr 2000, 20:02..