7. **Trigonometrical Functions
**

The common trig. functions are defined relative to a right-angled
triangle.

- tan
*x*= opposite/adjacent O/A - sin
*x*= opposite/hypotenuse O/H - cos
*x*= adjacent/hypotenuse A/H

In calculus, we need the angles measured in *radians* rather than degrees, with

so that 1 radian = 360° / 2π.

The common points used:

- 0° = 0 radians
- 90° = π/2 radians
- 180° = π radians
- 270° = 3π/2 radians
- 360° = 2π radians

For graphs of the common trig functions, see handout 2.

**Inverse Trig. Functions
**

If *y* = sin *x*, then *x* = sin^{-1}*y*
(also called arcsin *y*)

*y* = cos *x*, then *x* = cos^{-1}*y*
(also called arccos *y*)

*y* = tan *x*, then *x* = tan^{-1}*y*
(also called arctan *y*).

*Examples*

1. sin *x* = 0.32, *x* = sin^{-1}0.32 = 18.7 = 0.33 rads

2. tan *x* = 0.87, *x* = tan^{-1}0.87 = 41 = 0.71 rads

**Differentials of Trig functions
**

It can be shown that if

*y*= sin*x*, = cos*x**y*= cos*x*, = - sin*x**y*= tan*x*, =

We often meet compound functions in many applications and we can
use the Product, Chain and Sequential rules as before.

*Examples*

1. *y* = sin(*kx*)

= cos(*kx*).*k* = *k*cos
(*kx*) ** important**

2. *y* = 3cos *x *- 4sin(*x*^{2})

= -3sin *x* - 4 cos(*x*^{2}).2*x*

= -3sin *x* - 8*x*.cos(*x*^{2})

3. *y* = 7sin(5*x*^{2}) + 6ln {tan(5*x*)}

= 7cos(5*x*^{2}).10*x*
+ 5

= 70*x*.cos(5*x*^{2}) +

4. Ψ(*x*) = *A*sin , a typical
wave function for an electron in an orbital.

= *A*cos.

= .cos.

**8. Integration - Calculating Areas
**

In science, we often need to work out the **area** under a
graph. For example the work done to move a charged particle a
distance *r* through a potential difference *V* is given by
the area under the potential vs distance curve.

Another example is the total distance travelled by an object moving
at velocity *v*(*t*), (which is a function of time so
that the object is accelerating), is given by the area under the
*v*(*t*) vs *t* graph.

**Calculating Areas - counting squares
**

The most obvious way to calculate the area under a graph is to
draw it on graph paper and count the squares.

*Example* What is the area under the curve *y*(*x*)
= 2*x*^{2} from *x* = 1 to *x*
= 3?

a) Divide the area up into 2 trapezia

Area of trapezium = Area of rectangle + area of triangle

= (length x width) + (½ x length x height)

Area of **A** = (1 x 2) + (½ x 1 x 6) = 5 units

Area of **B** = (1 x 8) + (½ x 1 x 10) = 13 units

Total Area = **A**+**B** = 5+13 = __18 sq. units__

b) Now do it again with 4 trapezia

Area of **A** = (½ x 2) + (½ x ½ x 2½) = 15/8

Area of **B** = (½ x 4½) + (½ x ½ x 3½) = 33/8

Area of **C** = (½ x 8) + (½ x ½ x 4½) = 51/8

Area of **D** = (½ x 12½) + (½ x ½ x 5½) = 75/8

Total Area = **A**+**B**+**C**+**D** = **17.75
sq.units
**

You can see that as we divide up the area into smaller and smaller
strips, the approximation to the area gets better and better.
This is the basis for numerical solutions of areas (*e.g.*
Simpson's Rule - see textbooks).

The actual value of the area will be achieved when we have an
infinite number of strips, of width zero!

Needless to say we do not have to do this - there's a short cut
- *analytic integration
*

**Integration - Notation
**

The area under a curve, *y*(*x*), which has been divided
up into many strips of width *x* between the limits of *x*
= *a *and *x* = *b *is given by adding up all the
areas of the strips ( width x height).

Now, as *x* approaches 0 (*i.e*. the strips get vanishingly thin) we can replace
the Σ with an integration sign which is an extended S, for sum.

So we get:

The *dx* now serves to tell us what the name of the variable
is - we say *'with respect to x'
*

This process is called *integration
*

**Integration as the Reverse of Differentiation
**

We know that the differential of *y*(*x*) = *x*^{3}
is 3*x*^{2}. The process of reversing this,
whereby we generate a function from its derivative is *integration.*

*i.e.* = *y*(*x*), so that integration
of a differential gives the original function.

and [*y*(*x*).*dx*]
= *y*(*x*), so that differential of an integral gives the original function.

Compare other reversible functions such as:

*x*^{2}and- sin
*x*and sin^{-1}*x* - ln
*x*and e.^{x}

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