URL http://www.mast.queensu.ca/~helena/math006/week9.html

Math 006 - Introduction to Calculus

Lecture 9

New Stuff

We will cover the following topics in this class:

Main Part of class

We will find the derivatives of sine and cosine functions, and see them in some examples. We will cover some ideas from section 7.2, and examples from section 7.4, but we will not prove the results or cover things in quite as much detail as in the text book, and we will only look at inverses of trig functions (7.5) very briefly. (we will not cover other sections of this chapter, eg, we will not cover 7.1)

Definition of Trigonometric functions

Trigonometric functions are used for measuring angles. There are three main trigonometric functions, sine, cosine, and tangent, called sin, cos and tan for short; if the angle is x, write sin(x), cos(x), tan(x).

The following diagram describes how these are defined:

If you want a more gentle introduction to what trigonometry is for, see this page.

The inverse trigonometric functions tell you what angle has a certain sine, what value has a certain cosine, eg, what angle has sine(x) = 1/2 ? The answer is 60 degrees, so we say sin-1(1/2) = 60

The graphs of trigonometric functions.

To picture the graph of how sine(x) and cosine(x) change as the angle x changes, put the triangle in a circle:

The graphs of cosine and sine are periodic, so they are very useful for describing all kinds of periodic behaviour:

Derivatives of trigonometric functions

The derivative of trigonometric functions tells us the rate of change of the periodic behaviour that's being described.

You need to learn these rules for finding the derivatives:

d/dx(sin(x)) = cos(x)
d/dx(cos(x)) = -sin(x)

In the text book proofs of these formulae are given, but we are not going to cover that. There are also derivatives of other trigonometric functions

The important thing to note is that the derivatives of the trigonometric funtions are simply related to each other, and by examining the graphs, we can understand why this is.

Combining trigonometric functions with other functions

We will also cover some examples where trigonometric functions are combined with other functions, and see what these functions look like, and what are their derivatives, eg:

You can find the derivatives of these using product rule, chain rule, etc. Try them for yourself; check the answers by clicking on the equation. (as soon as I write the web page with the solutions to this!)

Rules for dealing with trig functions

We can do all the same kinds of problems for trigonometric functions as we did for polynomial functions, eg, find equations for tangents, or maximum and minimum values. But trigonometric functions are a little more difficult than polynomial functions, and there are a few problems that come up. The following are the main things that we'll use to solve these problems.

Angle formulae

Here are formulae you may need if you have an equation involving something like sin(x) and sin(2x) at the same time; this is a way to compare sine and cosine of different angles:

SIN(A + B) = SIN(A)*COS(B) + COS(A)*SIN(B)
COS(A + B) = COS(A)*COS(B) - SIN(A)*SIN(B)

Another even more important formula to remember, which comes straight from Pythagoras' theorem is

SIN2(A) + COS2(A) = 1

I will be putting up examples to show how these rules can be used in calculus soon.

Inverse of trigonometric functions

Inverses are about solving equations.

For example, if you want to solve

x2 = 4,
you take the sqaure root, and get

x = 41/2 = 2

Generally, if

x2 = a,
then
x = a1/2
So, the square root function gives you the solution to the square function. We describe this situation by saying that
the square root function is the inverse of the square function.

For any function f(x), the inverse is called f-1(x), this means that

If f(x) = A, then the solution is
x = f-1(A)

For example, if the function is

f(x) = x2 + 4x + 1 ?
Suppose we want to find when this function has a particular value, say 7. We want to solve
f(x) = x2 + 4x + 1 = 7

We rewrite this as

f(x) = x2 + 4x + 1-7 =

And then we can solve it using the quadratic formula:

x = (-4 + (16 - 4(1-7))1/2)/2

We can do the same thing for any number A,

If x2 + 4x + 1 = A
Then one possible solution is
x = (-4 + (16 - 4(1-A))1/2)/2

In inverse notation, we have

If f(x) = x2 + 4x + 1 = A
Then
f-1(A) = (-4 + (16 - 4(1-A))1/2)/2

What about the other solutions?

You might notice that in all these examples I am cheating, since I'm only giving you one possible solution; eg, for x2=4, x = 2 is a solution, but x = -2 is also a solution. But in this case, if you know one solution, it's easy to get the other solution - you just mulitply by -1.

Generally, if we have one solution, it would be nice if we could find other solutions from that one solution in an easy way, just like this multiplication by -1 is pretty easy, once you've got one solution.

How to find infinitely many solutions to sin(X) = A

You can find this from looking at the graph, and I'm going to write out a detailed explaination here later.

The idea is:

If sin(X) = A, then we also have
sin(2pi + X) = A, sin(4pi + X) = A, sin(6pi + X) = A, sin(8pi + X) = A,
sin(pi - X) = A, sin(3pi - X) = A, sin(5pi - X) = A, sin(7pi - X) = A,

If cos(X) = A, then we also have
sin(2pi + X) = A, sin(4pi + X) = A, sin(6pi + X) = A, sin(8pi + X) = A,
sin(2pi - X) = A, sin(4pi - X) = A, sin(6pi - X) = A, sin(8pi - X) = A,

This can be understood by examining the graphs above.

Homework

The following questions from the text book should be done for practice, but are not to be handed in for marking.

Assignment

The following questions from the text book should be handed in Wednesday 1 April

Sample Quiz

There will be a quiz next week, which will be similar to the following question:

Find the derivative of the function

f(x) = sin(x2 + x + 1)

Use this to find the equation for the tangent to f(x) at x = 0

See quiz page for the solution.

Handout

Please send corrections to spelling mistakes, and any other comments, to me.

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