• # Temporary trigonometry page

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Another one of my big pages - please leave alone till it's finished. :p Comments, suggestions, blah to generalebriety, cheers.

## Definitions

Using the Cartesian coordinate system, the circle C is centred at the origin O and has radius 1. P is any point on that circle and has coordinates (a, b), and is the angle between OP and the positive x-axis, measured anticlockwise. Then the following two basic trigonometrical functions are defined:

Furthermore, the following four functions are defined:

Clearly tan and sec are not defined at any values of θ for which cos θ = 0; similarly, cot θ and cosec θ are only defined when sin θ does not equal 0.

Note that the notation "csc θ" will be preferred to "cosec θ" in this article.

## Identities

### Basic identities

#### Properties of the functions

By considering θ and (-θ) in the definitions of sin and cos, it is easy to see that

that is, sin is an odd function, and cos is an even function. This makes tan, cot and csc odd functions, and sec an even function, i.e.

#### Basic relationships between the functions

It is obvious from our definitions of sin and cos, and by reflective symmetry, that

By dividing the first identity by the second, or the second by the first, respectively, we get

It is also immediately obvious, by taking the reciprocal of each of the first two identities, that

### Circular identities

From our definitions of sin and cos, and using Pythagoras' theorem, it is easily seen that

By dividing through by sin²x and cos²x respectively, we end up with the two identities given below. Although sin²x and cos²x can equal 0 (exactly when sin x = 0 and cos x = 0, respectively), our definitions of tan, sec, cosec and cot allow us to divide through.

[insert proof of sin(A+B) formula here :p]

By considering sin(A - B) = sin(A + (-B)):

By considering cos(A + B) = sin(π/2 - (A + B)) = sin((π/2 - A) - B):

By considering cos(A - B) = cos(A + (-B)):

It is immediately obvious that

which simplifies to the following identity when the numerator and denominator are divided by cos A and cos B:

Similarly,

We can derive the addition identities for cot in a similar way.

The addition identities for sec and csc are derived by taking the reciprocal of the identities for cos and sin, respectively, and multiplying the numerator and denominator by (sec A sec B csc A csc B).

#### Double and half angle formulae

The double angle formulae are derived simply by setting B = A, and are as follows:

The half angle formulae are equivalent to the double angle formulae, and are formed by replacing A by A/2 in the above formulae.

#### "t"-formulae

The half angle formula for sin can be written as follows, by multiplying the numerator and denominator by sec²θ.

The same can be done with the half angle formula for cos.

Then, by writing t = tan(x/2), we arrive at the following six identities:

The use of the substitution t = tan(x/2) is called t-substitution, and can be very useful in solving otherwise difficult or impossible integrals involving the simple trig functions.

### Factor formulae

#### Product formulae

We know that

and

Adding these two results in the product formula

Subtracting gives the formula

Similar manipulation of the addition identities for cos gives the two formulae

#### Sum formulae

The sum formulae are a direct result of the product formulae, by substituting A = P + Q, B = P - Q in the formulae

etc.

This gives:

## Calculus

### Differentiation

[insert proof that x --> 0 => sinx / x --> 1]

To differentiate sin x from first principles, let us consider the gradient m of the line between the points (x, sin x) and (x+o, sin(x+o)), where o is small.

As o tends to 0, [sin(o/2)]/(o/2) tends to 1, and cos(x + o/2) tends to cos x, and so m tends to cos x; that is,

Remember that cos x = sin(π/2 - x). Replacing x with (π/2 - x), and using the chain rule, in the above:

That is,

Using the quotient rule:

Deriving the final two by noting that cot x = tan(π/2 - x) and csc x = sec(π/2 - x) respectively:

### Integration

Arbitrary constants are not added here.

As a direct result of the differentiation results,

Also,

The other two functions can be integrated with a small amount of manipulation.

as the numerator is the derivative of the denominator.

as the numerator is the derivative of the denominator.