Identities

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Identities

You will also need to know the graphs and properties of the reciprocal functions:

Reciprocal functions

The following properties apply to any reciprocal function:

  1. The reciprocal of zero is +∞
  2. The reciprocal of +∞ is zero
  3. The reciprocal of 1 is 1
  4. The reciprocal of -1 is -1
  5. Where the function has a maximum value, its reciprocal has a minimum value
  6. If a function increases, the reciprocal decreases
  7. A function and its reciprocal have the same sign

The curves of cosec x, sec x and cot x are shown below:

GraphsGraph

From a right angled triangle we know that:

tan

cos2θ + sin2θ = 1

It can also be shown that:

1 + tan2θ = sec2θ and cot2θ + 1 = cosec2θ

(Try dividing the second expression by cos2θ to get the first rearrangement, and separately divide cos2θ + sin2θ = 1, by sin2θ to get the other formula.)

These are Trigonometric Identities and useful for rewriting equations so that they can be solved, integrated, simplified etc.

Trigonometric functions of angles like A + B and A − B can be expressed in terms of the trigonometric functions of A and B.

These are called compound angle identities:

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

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

Tan identity

Remember: take care with the signs when using these formulae.

The compound angle formulae can also be used with two equal angles i.e. A = B.

If we replace B with A in the compound angle formulae for (A + B), we have:

sin 2A = 2(sin A cos A)

cos 2A = cos2A - sin2A

Tan formula

Also,

cos 2A = cos 2A - sin 2A = 1 - 2sin2A = 2cos2A - 1

The use for this final rearrangement is when integrating cos2x or sin2x.

We use cos2 x = ½cos 2x + ½ and sin2 x = ½ - ½ cos 2x which we can integrate.

tan 2A

Using this double angle formula for tan 2A and the two identities:

sin 2A and cos 2A

We can replace 2A with x and use T for tan(x/2).

This gives us the following identities, which allow all the trigonometric functions of any angle to be expressed in terms of T.

Half angle formulae in terms of T

The formulae we have met so far involve manipulating single expressions of sin x and cos x. If we wish to add sin or cos expressions together we need to use the factor formulae, which are derived from the compound angle rules we met earlier.

The compound angle formulae can be combined to give:

2sin A cos B = sin (A + B) + sin (A − B)
2cos A sin B = sin (A + B) - sin (A − B)
2cos A cos B = cos (A + B) + cos (A − B)
−2sin A sin B = cos (A + B) - cos (A − B)

If we simplify the right hand side of each of these equations by substituting

A + B = J and A − B = K, we create the factor formulae:

Factor formulae

The factor formulae allow us to add and subtract expressions that are all sines or all cosines. If we wish to add a sine and a cosine expression together we have to use a different method.

This method is based upon the fact that combining a sine and a cosine will generate another cos curve with a greater amplitude and which is a number of degrees out of phase with the graph of cos θ.

This means that it can be written as R cos(θ - α), where R represents the amplitude and α represents the number of degrees the graph is out of phase (to the right).

The solution is based upon the expansion of cos(θ - α).

Example:

Write 5 sin x + 12 cos x in the form R cos (θ - α)

R cos (θ - α) = R (cos θ cos α + sin θ sin α)

By matching this expansion to the question we get:

R cos θ cos α = 12 cos θ and R sin θ sin α = 5 sin θ

This gives:

R cos α = 12 and R sin α = 5

By illustrating this with a right-angled triangle, we get,

Triangle

Therefore: α = 22.6o

Therefore: 5 sin θ + 12 cos θ = 13 cos(θ - 22.6)

It has a maximum value of 13 and is 22.6o out of phase with the graph of cos θ.

Note: This procedure would work with Rsin(θ + α).

Check to see if you can get a similar answer - it should be 13 sin (θ + 67.4)

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